id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
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proofwiki-11100 | Burnout Height of Upward Rocket under Constant Gravity | Let $R$ be a rocket whose structural mass is $m_1$.
Let $R$ contain fuel of initial mass $m_2$.
Let $R$ be fired straight up from the surface of a planet whose gravitational field exerts an Acceleration Due to Gravity of $g$, assumed constant.
Let $R$ burn fuel at a constant rate $a$, with a constant exhaust velocity $... | {{Proofread}}
{{MissingLinks}}
The total mass of the rocket at time $t$ is given by
:$m = m_1 + m_2 - at$
The time $t_b$ at which the rocket runs out of fuel is given by:
:$t_b = \dfrac {m_2} a$
Now we can calculate the total force on the rocket to be:
:$F = b a - mg$
where:
:$b a$ is the rocket's thrust as calculated ... | Let $R$ be a [[Definition:Rocket|rocket]] whose structural [[Definition:Mass|mass]] is $m_1$.
Let $R$ contain fuel of initial [[Definition:Mass|mass]] $m_2$.
Let $R$ be fired straight up from the surface of a [[Definition:Planet|planet]] whose [[Definition:Gravitational Field|gravitational field]] exerts an [[Acceler... | {{Proofread}}
{{MissingLinks}}
The total mass of the rocket at time $t$ is given by
:$m = m_1 + m_2 - at$
The time $t_b$ at which the rocket runs out of fuel is given by:
:$t_b = \dfrac {m_2} a$
Now we can calculate the total force on the rocket to be:
:$F = b a - mg$
where:
:$b a$ is the rocket's thrust as calcul... | Burnout Height of Upward Rocket under Constant Gravity | https://proofwiki.org/wiki/Burnout_Height_of_Upward_Rocket_under_Constant_Gravity | https://proofwiki.org/wiki/Burnout_Height_of_Upward_Rocket_under_Constant_Gravity | [
"Rocket Science"
] | [
"Definition:Rocket",
"Definition:Mass",
"Definition:Mass",
"Definition:Planet",
"Definition:Gravitational Field",
"Acceleration Due to Gravity",
"Definition:Rate",
"Definition:Exhaust Velocity",
"Definition:Gravitational Field",
"Definition:Burnout Height"
] | [
"Newton's Laws of Motion/Second Law",
"Newton's Laws of Motion/First Law"
] |
proofwiki-11101 | Linear First Order ODE/y' = x + y | The linear first order ODE:
:$(1): \quad \dfrac {\d y} {\d x} = x + y$
has the general solution:
:$y = C e^x - x - 1$ | Rearranging $(1)$:
:$(2): \quad \dfrac {\d y} {\d x} - y = x$
$(2)$ is a linear first order ODE in the form:
:$\dfrac {\d y} {\d x} + \map P x y = \map Q x$
where:
:$\map P x = -1$
:$\map Q x = x$
Thus:
{{begin-eqn}}
{{eqn | l = \int \map P x \rd x
| r = -\int \rd x
| c =
}}
{{eqn | r = - x
| c =
}}... | The [[Definition:Linear First Order ODE|linear first order ODE]]:
:$(1): \quad \dfrac {\d y} {\d x} = x + y$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$y = C e^x - x - 1$ | Rearranging $(1)$:
:$(2): \quad \dfrac {\d y} {\d x} - y = x$
$(2)$ is a [[Definition:Linear First Order ODE|linear first order ODE]] in the form:
:$\dfrac {\d y} {\d x} + \map P x y = \map Q x$
where:
:$\map P x = -1$
:$\map Q x = x$
Thus:
{{begin-eqn}}
{{eqn | l = \int \map P x \rd x
| r = -\int \rd x
... | Linear First Order ODE/y' = x + y | https://proofwiki.org/wiki/Linear_First_Order_ODE/y'_=_x_+_y | https://proofwiki.org/wiki/Linear_First_Order_ODE/y'_=_x_+_y | [
"Examples of Linear First Order ODEs"
] | [
"Definition:Linear First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Linear First Order Ordinary Differential Equation",
"Solution to Linear First Order Ordinary Differential Equation/Solution by Integrating Factor",
"Primitive of x by Exponential of a x"
] |
proofwiki-11102 | Linear First Order ODE/y' = x + y/y(0) = 1 | The linear first order ODE:
:$(1): \quad \dfrac {\d y} {\d x} = x + y$
with initial condition:
:$\map y 0 = 1$
has the particular solution:
:$y = 2 e^x - x - 1$ | From Linear First Order ODE: $y' = x + y$, the general solution of $(1)$ is:
:$y = C e^x - x - 1$
Setting $y = 1$ when $x = 0$ gives:
:$1 = C + 1$
from which $C = 2$.
Hence the result.
{{qed}} | The [[Definition:Linear First Order ODE|linear first order ODE]]:
:$(1): \quad \dfrac {\d y} {\d x} = x + y$
with [[Definition:Initial Condition|initial condition]]:
:$\map y 0 = 1$
has the [[Definition:Particular Solution to Differential Equation|particular solution]]:
:$y = 2 e^x - x - 1$ | From [[Linear First Order ODE/y' = x + y|Linear First Order ODE: $y' = x + y$]], the [[Definition:General Solution to Differential Equation|general solution]] of $(1)$ is:
:$y = C e^x - x - 1$
Setting $y = 1$ when $x = 0$ gives:
:$1 = C + 1$
from which $C = 2$.
Hence the result.
{{qed}} | Linear First Order ODE/y' = x + y/y(0) = 1 | https://proofwiki.org/wiki/Linear_First_Order_ODE/y'_=_x_+_y/y(0)_=_1 | https://proofwiki.org/wiki/Linear_First_Order_ODE/y'_=_x_+_y/y(0)_=_1 | [
"Examples of Linear First Order ODEs"
] | [
"Definition:Linear First Order Ordinary Differential Equation",
"Definition:Initial Condition",
"Definition:Differential Equation/Solution/Particular Solution"
] | [
"Linear First Order ODE/y' = x + y",
"Definition:Differential Equation/Solution/General Solution"
] |
proofwiki-11103 | Existence and Uniqueness of Solution for Linear Second Order ODE with two Initial Conditions | Let $\map P x$, $\map Q x$ and $\map R x$ be continuous real functions on a closed real interval $\closedint a b$.
Let $x_0$ be any point in $\closedint a b$.
Let $y_0$ and ${y_0}'$ be real numbers.
Then the linear second order ordinary differential equation:
:$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d ... | Let $z = \dfrac {\d y} {\d x}$.
Then a solution to $(1)$ will yield a particular solution to:
:$(2): \quad \begin{cases}
\dfrac {\d y} {\d x} = z & , \map y {x_0} = y_0 \\
& \\
\dfrac {\d z} {\d x} = -\map P x \dfrac {\d y} {\d x} - \map Q x y + \map R x & , \map z {x_0} = {y_0}'
\end{cases}$
From Lipschitz Condition o... | Let $\map P x$, $\map Q x$ and $\map R x$ be [[Definition:Continuous Real Function|continuous real functions]] on a [[Definition:Closed Real Interval|closed real interval]] $\closedint a b$.
Let $x_0$ be any point in $\closedint a b$.
Let $y_0$ and ${y_0}'$ be [[Definition:Real Number|real numbers]].
Then the [[Def... | Let $z = \dfrac {\d y} {\d x}$.
Then a [[Definition:Solution to Differential Equation|solution]] to $(1)$ will yield a [[Definition:Particular Solution to Differential Equation|particular solution]] to:
:$(2): \quad \begin{cases}
\dfrac {\d y} {\d x} = z & , \map y {x_0} = y_0 \\
& \\
\dfrac {\d z} {\d x} = -\map P ... | Existence and Uniqueness of Solution for Linear Second Order ODE with two Initial Conditions/Proof | https://proofwiki.org/wiki/Existence_and_Uniqueness_of_Solution_for_Linear_Second_Order_ODE_with_two_Initial_Conditions | https://proofwiki.org/wiki/Existence_and_Uniqueness_of_Solution_for_Linear_Second_Order_ODE_with_two_Initial_Conditions/Proof | [
"Linear Second Order ODEs"
] | [
"Definition:Continuous Real Function",
"Definition:Real Interval/Closed",
"Definition:Real Number",
"Definition:Linear Second Order Ordinary Differential Equation",
"Definition:Unique",
"Definition:Differential Equation/Solution/Particular Solution"
] | [
"Definition:Differential Equation/Solution",
"Definition:Differential Equation/Solution/Particular Solution",
"Lipschitz Condition on Linear ODE of Continuous Functions",
"Definition:Lipschitz Continuity",
"Picard's Existence Theorem"
] |
proofwiki-11104 | General Solution of Linear 2nd Order ODE from Homogeneous 2nd Order ODE and Particular Solution | Consider the nonhomogeneous linear second order ODE:
:$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = \map R x$
Let $\map {y_g} x$ be the general solution of the homogeneous linear second order ODE:
:$(2): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = ... | Let $\map {y_g} {x, C_1, C_2}$ be a general solution of $(2)$.
Note that $C_1$ and $C_2$ are the two arbitrary constants that are to be expected of a second order ODE.
Let $\map {y_p} x$ be a certain fixed particular solution of $(1)$.
Let $\map y x$ be an arbitrary particular solution of $(1)$.
Then:
{{begin-eqn}}
{{e... | Consider the [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]]:
:$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = \map R x$
Let $\map {y_g} x$ be the [[Definition:General Solution to Differential Equation|general solution]] of the [[Definiti... | Let $\map {y_g} {x, C_1, C_2}$ be a [[Definition:General Solution to Differential Equation|general solution]] of $(2)$.
Note that $C_1$ and $C_2$ are the two [[Definition:Arbitrary Constant|arbitrary constants]] that are to be expected of a [[Definition:Second Order ODE|second order ODE]].
Let $\map {y_p} x$ be a cer... | General Solution of Linear 2nd Order ODE from Homogeneous 2nd Order ODE and Particular Solution | https://proofwiki.org/wiki/General_Solution_of_Linear_2nd_Order_ODE_from_Homogeneous_2nd_Order_ODE_and_Particular_Solution | https://proofwiki.org/wiki/General_Solution_of_Linear_2nd_Order_ODE_from_Homogeneous_2nd_Order_ODE_and_Particular_Solution | [
"Linear Second Order ODEs"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Homogeneous Linear Second Order ODE",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Differential Equation/Solution/General Solution",
"Definition:Arbitrary Constant",
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Differentia... |
proofwiki-11105 | Trivial Solution of Homogeneous Linear 2nd Order ODE | The homogeneous linear second order ODE:
:$\dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$
has the particular solution:
:$\map y x = 0$
that is, the zero constant function.
This particular solution is referred to as the '''trivial solution'''. | We have:
:$\map {\dfrac {\d} {\d x} } 0 = 0$
and so:
:$\map {\dfrac {\d^2} {\d x^2} } 0 = 0$
from which:
:$\dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$
Hence the result.
{{Qed}} | The [[Definition:Homogeneous Linear Second Order ODE|homogeneous linear second order ODE]]:
:$\dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$
has the [[Definition:Particular Solution|particular solution]]:
:$\map y x = 0$
that is, the [[Definition:Constant Mapping|zero constant function]].
... | We have:
:$\map {\dfrac {\d} {\d x} } 0 = 0$
and so:
:$\map {\dfrac {\d^2} {\d x^2} } 0 = 0$
from which:
:$\dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$
Hence the result.
{{Qed}} | Trivial Solution of Homogeneous Linear 2nd Order ODE | https://proofwiki.org/wiki/Trivial_Solution_of_Homogeneous_Linear_2nd_Order_ODE | https://proofwiki.org/wiki/Trivial_Solution_of_Homogeneous_Linear_2nd_Order_ODE | [
"Homogeneous LSOODEs"
] | [
"Definition:Homogeneous Linear Second Order ODE",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Constant Mapping",
"Definition:Differential Equation/Solution/Particular Solution"
] | [] |
proofwiki-11106 | Linear Combination of Solutions to Homogeneous Linear 2nd Order ODE | Let $c_1$ and $c_2$ be real numbers.
Let $\map {y_1} x$ and $\map {y_2} x$ be particular solutions to the homogeneous linear second order ODE:
:$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$
Then:
:$c_1 \, \map {y_1} x + c_2 \, \map {y_2} x$
is also a particular solution to $(1)$... | {{begin-eqn}}
{{eqn | o =
| r = \paren {c_1 \, \map {y_1} x + c_2 \, \map {y_2} x}' ' + \map P x \paren {c_1 \, \map {y_1} x + c_2 \, \map {y_2} x}' + \map Q x \paren {c_1 \, \map {y_1} x + c_2 \, \map {y_2} x}
| c =
}}
{{eqn | r = \paren {c_1 \, \map {y_1' '} x + c_2 \, \map {y_2' '} x} + \map P x \paren... | Let $c_1$ and $c_2$ be [[Definition:Real Number|real numbers]].
Let $\map {y_1} x$ and $\map {y_2} x$ be [[Definition:Particular Solution|particular solutions]] to the [[Definition:Homogeneous Linear Second Order ODE|homogeneous linear second order ODE]]:
:$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} ... | {{begin-eqn}}
{{eqn | o =
| r = \paren {c_1 \, \map {y_1} x + c_2 \, \map {y_2} x}' ' + \map P x \paren {c_1 \, \map {y_1} x + c_2 \, \map {y_2} x}' + \map Q x \paren {c_1 \, \map {y_1} x + c_2 \, \map {y_2} x}
| c =
}}
{{eqn | r = \paren {c_1 \, \map {y_1' '} x + c_2 \, \map {y_2' '} x} + \map P x \paren... | Linear Combination of Solutions to Homogeneous Linear 2nd Order ODE | https://proofwiki.org/wiki/Linear_Combination_of_Solutions_to_Homogeneous_Linear_2nd_Order_ODE | https://proofwiki.org/wiki/Linear_Combination_of_Solutions_to_Homogeneous_Linear_2nd_Order_ODE | [
"Homogeneous LSOODEs"
] | [
"Definition:Real Number",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Homogeneous Linear Second Order ODE",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Linear Combination of Solutions to Homogeneous Linear 2nd Order ODE",
"Definition:Differ... | [
"Linear Combination of Derivatives"
] |
proofwiki-11107 | Solution of Linear 2nd Order ODE Tangent to X-Axis | Let $\map {y_p} x$ be a particular solution to the homogeneous linear second order ODE:
:$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$
on a closed interval $\closedint a b$.
Let there exist $\xi \in \closedint a b$ such that the curve in the cartesian plane described by $y = \ma... | {{AimForCont}} $y_p$ is not the zero constant function.
From Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another, there exists another particular solution to $(1)$ such that $y_1$ and $y_2$ are linearly independent.
At the point $\xi$:
:$\map {y_p} \xi = 0$
:$\map { {y_p}'} \xi = 0$
Taking ... | Let $\map {y_p} x$ be a [[Definition:Particular Solution|particular solution]] to the [[Definition:Homogeneous Linear Second Order ODE|homogeneous linear second order ODE]]:
:$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$
on a [[Definition:Closed Real Interval|closed interval]] $... | {{AimForCont}} $y_p$ is not the [[Definition:Constant Mapping|zero constant function]].
From [[Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another]], there exists another [[Definition:Particular Solution|particular solution]] to $(1)$ such that $y_1$ and $y_2$ are [[Definition:Linearly Ind... | Solution of Linear 2nd Order ODE Tangent to X-Axis | https://proofwiki.org/wiki/Solution_of_Linear_2nd_Order_ODE_Tangent_to_X-Axis | https://proofwiki.org/wiki/Solution_of_Linear_2nd_Order_ODE_Tangent_to_X-Axis | [
"Linear Second Order ODEs"
] | [
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Homogeneous Linear Second Order ODE",
"Definition:Real Interval/Closed",
"Definition:Line/Curve",
"Definition:Cartesian Plane",
"Definition:Tangent Line",
"Definition:Axis/X-Axis",
"Definition:Constant Mapping"
] | [
"Definition:Constant Mapping",
"Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Linearly Independent Real Functions",
"Definition:Wronskian",
"Definition:Zero (Number)",
"Zero Wronskian of Sol... |
proofwiki-11108 | Solutions of Linear 2nd Order ODE have Common Zero iff Linearly Dependent | Let $\map {y_1} x$ and $\map {y_2} x$ be particular solutions to the homogeneous linear second order ODE:
:$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$
on a closed interval $\closedint a b$.
Let $y_1$ and $y_2$ both have a zero for the same value of $x$ in $\closedint a b$.
The... | Let $\xi \in \closedint a b$ be such that $\map {y_1} \xi = \map {y_2} \xi = 0$.
Consider the Wronskian $\map W {y_1, y_2}$ at $\xi$:
{{begin-eqn}}
{{eqn | l = \map W {\map {y_1} \xi, \map {y_2} \xi}
| r = \map {y_1} \xi \map { {y_2}'} \xi - \map {y_2} \xi \map { {y_1}'} \xi
| c =
}}
{{eqn | r = 0 \cdot \m... | Let $\map {y_1} x$ and $\map {y_2} x$ be [[Definition:Particular Solution|particular solutions]] to the [[Definition:Homogeneous Linear Second Order ODE|homogeneous linear second order ODE]]:
:$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$
on a [[Definition:Closed Real Interval|c... | Let $\xi \in \closedint a b$ be such that $\map {y_1} \xi = \map {y_2} \xi = 0$.
Consider the [[Definition:Wronskian|Wronskian]] $\map W {y_1, y_2}$ at $\xi$:
{{begin-eqn}}
{{eqn | l = \map W {\map {y_1} \xi, \map {y_2} \xi}
| r = \map {y_1} \xi \map { {y_2}'} \xi - \map {y_2} \xi \map { {y_1}'} \xi
| c ... | Solutions of Linear 2nd Order ODE have Common Zero iff Linearly Dependent | https://proofwiki.org/wiki/Solutions_of_Linear_2nd_Order_ODE_have_Common_Zero_iff_Linearly_Dependent | https://proofwiki.org/wiki/Solutions_of_Linear_2nd_Order_ODE_have_Common_Zero_iff_Linearly_Dependent | [
"Linear Second Order ODEs"
] | [
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Homogeneous Linear Second Order ODE",
"Definition:Real Interval/Closed",
"Definition:Root of Mapping",
"Definition:Constant",
"Definition:Multiple",
"Definition:Linearly Dependent Real Functions"
] | [
"Definition:Wronskian",
"Zero Wronskian of Solutions of Homogeneous Linear Second Order ODE",
"Zero Wronskian of Solutions of Homogeneous Linear Second Order ODE iff Linearly Dependent",
"Definition:Linearly Dependent Real Functions"
] |
proofwiki-11109 | Real Function is Linearly Dependent with Zero Function | Let $\map f x$ be a real function defined on a closed interval $\closedint a b$.
Let $\map g x$ be the constant zero function on $\closedint a b$:
:$\forall x \in \closedint a b: \map g x = 0$
Then $f$ and $g$ are linearly dependent on $\closedint a b$. | We have that:
:$\forall x \in \closedint a b: \map g x = 0 = 0 \times \map f x$
and $0 \in \R$.
Hence the result by definition of linearly dependent real functions. | Let $\map f x$ be a [[Definition:Real Function|real function]] defined on a [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Let $\map g x$ be the [[Definition:Constant Mapping|constant zero function]] on $\closedint a b$:
:$\forall x \in \closedint a b: \map g x = 0$
Then $f$ and $g$ are [[Defi... | We have that:
:$\forall x \in \closedint a b: \map g x = 0 = 0 \times \map f x$
and $0 \in \R$.
Hence the result by definition of [[Definition:Linearly Dependent Real Functions|linearly dependent real functions]]. | Real Function is Linearly Dependent with Zero Function | https://proofwiki.org/wiki/Real_Function_is_Linearly_Dependent_with_Zero_Function | https://proofwiki.org/wiki/Real_Function_is_Linearly_Dependent_with_Zero_Function | [
"Real Analysis"
] | [
"Definition:Real Function",
"Definition:Real Interval/Closed",
"Definition:Constant Mapping",
"Definition:Linearly Dependent Real Functions"
] | [
"Definition:Linearly Dependent Real Functions"
] |
proofwiki-11110 | Two Linearly Independent Solutions of Homogeneous Linear Second Order ODE generate General Solution | Let $\map {y_1} x$ and $\map {y_2} x$ be particular solutions to the homogeneous linear second order ODE:
:$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$
on a closed interval $\closedint a b$.
Let $y_1$ and $y_2$ be linearly independent.
Then the general solution to $(1)$ is:
:$y... | Let $\map y x$ be any particular solution to $(1)$ on $\closedint a b$.
It is to be shown that constants $C_1$ and $C_2$ can be found such that:
:$\map y x = C_1 \map {y_1} x + C_2 \map {y_2} x$
for all $x \in \closedint a b$.
By Existence and Uniqueness of Solution for Linear Second Order ODE with two Initial Conditio... | Let $\map {y_1} x$ and $\map {y_2} x$ be [[Definition:Particular Solution|particular solutions]] to the [[Definition:Homogeneous Linear Second Order ODE|homogeneous linear second order ODE]]:
:$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$
on a [[Definition:Closed Real Interval|c... | Let $\map y x$ be any [[Definition:Particular Solution|particular solution]] to $(1)$ on $\closedint a b$.
It is to be shown that [[Definition:Constant|constants]] $C_1$ and $C_2$ can be found such that:
:$\map y x = C_1 \map {y_1} x + C_2 \map {y_2} x$
for all $x \in \closedint a b$.
By [[Existence and Uniqueness of... | Two Linearly Independent Solutions of Homogeneous Linear Second Order ODE generate General Solution | https://proofwiki.org/wiki/Two_Linearly_Independent_Solutions_of_Homogeneous_Linear_Second_Order_ODE_generate_General_Solution | https://proofwiki.org/wiki/Two_Linearly_Independent_Solutions_of_Homogeneous_Linear_Second_Order_ODE_generate_General_Solution | [
"Homogeneous LSOODEs"
] | [
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Homogeneous Linear Second Order ODE",
"Definition:Real Interval/Closed",
"Definition:Linearly Independent Real Functions",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Arbitrary Constant"
] | [
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Constant",
"Existence and Uniqueness of Solution for Linear Second Order ODE with two Initial Conditions",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Image (Set Theory)/Mapping/Element",
"Definit... |
proofwiki-11111 | Zero Wronskian of Solutions of Homogeneous Linear Second Order ODE | Let $\map {y_1} x$ and $\map {y_2} x$ be particular solutions to the homogeneous linear second order ODE:
:$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$
on a closed interval $\closedint a b$.
Let $y_1$ and $y_2$ be linearly independent.
Then their Wronskian is either never zero,... | {{begin-eqn}}
{{eqn | l = \map W {y_1, y_2}
| r = y_1 {y_2}' - y_2 {y_1}'
| c =
}}
{{eqn | ll= \leadsto
| l = \map {W'} {y_1, y_2}
| r = \paren {y_1 {y_2}' ' + {y_1}' {y_2}'} - \paren {y_2 {y_1}' ' + {y_2}' {y_1}'}
| c = Product Rule for Derivatives
}}
{{eqn | r = y_1 {y_2}' ' - y_2 {y_1}... | Let $\map {y_1} x$ and $\map {y_2} x$ be [[Definition:Particular Solution|particular solutions]] to the [[Definition:Homogeneous Linear Second Order ODE|homogeneous linear second order ODE]]:
:$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$
on a [[Definition:Closed Real Interval|c... | {{begin-eqn}}
{{eqn | l = \map W {y_1, y_2}
| r = y_1 {y_2}' - y_2 {y_1}'
| c =
}}
{{eqn | ll= \leadsto
| l = \map {W'} {y_1, y_2}
| r = \paren {y_1 {y_2}' ' + {y_1}' {y_2}'} - \paren {y_2 {y_1}' ' + {y_2}' {y_1}'}
| c = [[Product Rule for Derivatives]]
}}
{{eqn | r = y_1 {y_2}' ' - y_2 {... | Zero Wronskian of Solutions of Homogeneous Linear Second Order ODE | https://proofwiki.org/wiki/Zero_Wronskian_of_Solutions_of_Homogeneous_Linear_Second_Order_ODE | https://proofwiki.org/wiki/Zero_Wronskian_of_Solutions_of_Homogeneous_Linear_Second_Order_ODE | [
"Homogeneous LSOODEs",
"Wronskians"
] | [
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Homogeneous Linear Second Order ODE",
"Definition:Real Interval/Closed",
"Definition:Linearly Independent Real Functions",
"Definition:Wronskian"
] | [
"Product Rule for Derivatives",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Linear First Order Ordinary Differential Equation",
"Solution to Linear First Order Ordinary Differential Equation",
"Definition:Exponential Function/Real"
] |
proofwiki-11112 | Real Number Line is Lindelöf | Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Then $\struct {\R, \tau_d}$ is a Lindelöf Space. | From Real Number Line is Second-Countable we have that $\struct {\R, \tau_d}$ is a second-countable space.
The result follows from Second-Countable Space is Lindelöf.
{{qed}} | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Then $\struct {\R, \tau_d}$ is a [[Definition:Lindelöf Space|Lindelöf Space]]. | From [[Real Number Line is Second-Countable]] we have that $\struct {\R, \tau_d}$ is a [[Definition:Second-Countable Space|second-countable space]].
The result follows from [[Second-Countable Space is Lindelöf]].
{{qed}} | Real Number Line is Lindelöf/Proof 1 | https://proofwiki.org/wiki/Real_Number_Line_is_Lindelöf | https://proofwiki.org/wiki/Real_Number_Line_is_Lindelöf/Proof_1 | [
"Real Number Line is Lindelöf",
"Real Number Line with Euclidean Topology",
"Examples of Lindelöf Spaces"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Lindelöf Space"
] | [
"Real Number Line is Second-Countable",
"Definition:Second-Countable Space",
"Second-Countable Space is Lindelöf"
] |
proofwiki-11113 | Real Number Line is Lindelöf | Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Then $\struct {\R, \tau_d}$ is a Lindelöf Space. | From Real Number Line is $\sigma$-Compact we have that $\struct {\R, \tau_d}$ is a $\sigma$-compact space.
The result follows from $\sigma$-Compact Space is Lindelöf.
{{qed}} | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Then $\struct {\R, \tau_d}$ is a [[Definition:Lindelöf Space|Lindelöf Space]]. | From [[Real Number Line is Sigma-Compact|Real Number Line is $\sigma$-Compact]] we have that $\struct {\R, \tau_d}$ is a [[Definition:Sigma-Compact Space|$\sigma$-compact space]].
The result follows from [[Sigma-Compact Space is Lindelöf|$\sigma$-Compact Space is Lindelöf]].
{{qed}} | Real Number Line is Lindelöf/Proof 2 | https://proofwiki.org/wiki/Real_Number_Line_is_Lindelöf | https://proofwiki.org/wiki/Real_Number_Line_is_Lindelöf/Proof_2 | [
"Real Number Line is Lindelöf",
"Real Number Line with Euclidean Topology",
"Examples of Lindelöf Spaces"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Lindelöf Space"
] | [
"Real Number Line is Sigma-Compact",
"Definition:Sigma-Compact Space",
"Sigma-Compact Space is Lindelöf"
] |
proofwiki-11114 | Zero Wronskian of Solutions of Homogeneous Linear Second Order ODE iff Linearly Dependent | Let $\map {y_1} x$ and $\map {y_2} x$ be particular solutions to the homogeneous linear second order ODE:
:$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$
on a closed interval $\closedint a b$.
Then:
:$y_1$ and $y_2$ are linearly dependent
{{iff}}:
:the Wronskian $\map W {y_1, y_2... | === Sufficient Condition ===
Let $y_1$ and $y_2$ are linearly dependent.
Suppose either $y_1$ or $y_2$ is zero everywhere on $\closedint a b$.
Then either ${y_1}'$ or ${y_2}'$ is also zero everywhere on $\closedint a b$.
Thus:
:$y_1 {y_2}' - y_2 {y_1}' = 0$
and so by definition $\map W {y_1, y_2} = 0$ everywhere on $\c... | Let $\map {y_1} x$ and $\map {y_2} x$ be [[Definition:Particular Solution|particular solutions]] to the [[Definition:Homogeneous Linear Second Order ODE|homogeneous linear second order ODE]]:
:$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$
on a [[Definition:Closed Real Interval|c... | === Sufficient Condition ===
Let $y_1$ and $y_2$ are [[Definition:Linearly Dependent Real Functions|linearly dependent]].
Suppose either $y_1$ or $y_2$ is zero everywhere on $\closedint a b$.
Then either ${y_1}'$ or ${y_2}'$ is also zero everywhere on $\closedint a b$.
Thus:
:$y_1 {y_2}' - y_2 {y_1}' = 0$
and so by... | Zero Wronskian of Solutions of Homogeneous Linear Second Order ODE iff Linearly Dependent | https://proofwiki.org/wiki/Zero_Wronskian_of_Solutions_of_Homogeneous_Linear_Second_Order_ODE_iff_Linearly_Dependent | https://proofwiki.org/wiki/Zero_Wronskian_of_Solutions_of_Homogeneous_Linear_Second_Order_ODE_iff_Linearly_Dependent | [
"Homogeneous LSOODEs"
] | [
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Homogeneous Linear Second Order ODE",
"Definition:Real Interval/Closed",
"Definition:Linearly Dependent Real Functions",
"Definition:Wronskian"
] | [
"Definition:Linearly Dependent Real Functions",
"Definition:Linearly Dependent Real Functions",
"Derivative of Constant Multiple",
"Definition:Linearly Dependent Real Functions"
] |
proofwiki-11115 | Linear Second Order ODE/y'' + y = 0 | The second order ODE:
:$(1): \quad y' ' + y = 0$
has the general solution:
:$y = C_1 \sin x + C_2 \cos x$ | Using Solution of Second Order Differential Equation with Missing Independent Variable, $(1)$ can be expressed as:
:$p \dfrac {\d p} {\d y} = -y$
where $p = \dfrac {\d y} {\d x}$.
From:
:First Order ODE: $y \rd y = k x \rd x$
with $k = 1$, this has the solution:
:$p^2 = -y^2 + C$
or:
:$p^2 + y^2 = C$
As the {{LHS}} is... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' + y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 \sin x + C_2 \cos x$ | Using [[Solution of Second Order Differential Equation with Missing Independent Variable]], $(1)$ can be expressed as:
:$p \dfrac {\d p} {\d y} = -y$
where $p = \dfrac {\d y} {\d x}$.
From:
:[[First Order ODE/y dy = k x dx|First Order ODE: $y \rd y = k x \rd x$]]
with $k = 1$, this has the [[Definition:General Soluti... | Linear Second Order ODE/y'' + y = 0/Proof 1 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_y_=_0/Proof_1 | [
"Linear Second Order ODE/y'' + y = 0",
"Examples of Linear Second Order ODE/y'' + k^2 y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Solution of Second Order Differential Equation with Missing Independent Variable",
"First Order ODE/y dy = k x dx",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Square/Function",
"Definition:Positive/Real Number",
"Solution to Separable Differential Equation",
"Primitive of... |
proofwiki-11116 | Linear Second Order ODE/y'' + y = 0 | The second order ODE:
:$(1): \quad y' ' + y = 0$
has the general solution:
:$y = C_1 \sin x + C_2 \cos x$ | $(1)$ can be seen to be a special case of:
:$(2): \quad$ Linear Second Order ODE: $y' ' + k^2 y = 0$
with $k = 1$.
$(2)$ has the solution:
:$y = C_1 \sin k x + C_2 \cos k x$
Hence setting $k = 1$:
:$y = C_1 \sin x + C_2 \cos x$
{{qed}} | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' + y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 \sin x + C_2 \cos x$ | $(1)$ can be seen to be a special case of:
:$(2): \quad$ [[Linear Second Order ODE/y'' + k^2 y = 0|Linear Second Order ODE: $y' ' + k^2 y = 0$]]
with $k = 1$.
$(2)$ has the solution:
:$y = C_1 \sin k x + C_2 \cos k x$
Hence setting $k = 1$:
:$y = C_1 \sin x + C_2 \cos x$
{{qed}} | Linear Second Order ODE/y'' + y = 0/Proof 2 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_y_=_0/Proof_2 | [
"Linear Second Order ODE/y'' + y = 0",
"Examples of Linear Second Order ODE/y'' + k^2 y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Linear Second Order ODE/y'' + k^2 y = 0"
] |
proofwiki-11117 | Linear Second Order ODE/y'' + y = 0 | The second order ODE:
:$(1): \quad y' ' + y = 0$
has the general solution:
:$y = C_1 \sin x + C_2 \cos x$ | We have that:
{{begin-eqn}}
{{eqn | l = \frac \d {\d x} \, \sin x
| r = \cos x
| c = Derivative of Sine Function
}}
{{eqn | l = \frac \d {\d x} \, \cos x
| r = -\sin x
| c = Derivative of Cosine Function
}}
{{eqn | ll= \leadsto
| l = \frac {\d^2} {\d x^2} \, \cos x
| r = -\cos x
... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' + y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 \sin x + C_2 \cos x$ | We have that:
{{begin-eqn}}
{{eqn | l = \frac \d {\d x} \, \sin x
| r = \cos x
| c = [[Derivative of Sine Function]]
}}
{{eqn | l = \frac \d {\d x} \, \cos x
| r = -\sin x
| c = [[Derivative of Cosine Function]]
}}
{{eqn | ll= \leadsto
| l = \frac {\d^2} {\d x^2} \, \cos x
| r = -\co... | Linear Second Order ODE/y'' + y = 0/Proof 3 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_y_=_0/Proof_3 | [
"Linear Second Order ODE/y'' + y = 0",
"Examples of Linear Second Order ODE/y'' + k^2 y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Derivative of Sine Function",
"Derivative of Cosine Function",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Constant Mapping",
"Definition:Real Interval/Closed",
"Definition:Wronskian",
"Sum of Squares of Sine and Cosine",
"Definition:Wronskian",
"Definition:Homogene... |
proofwiki-11118 | Linear Second Order ODE/y'' + y = 0 | The second order ODE:
:$(1): \quad y' ' + y = 0$
has the general solution:
:$y = C_1 \sin x + C_2 \cos x$ | Note that:
{{begin-eqn}}
{{eqn | l = y_1
| r = \sin x
| c =
}}
{{eqn | ll= \leadsto
| l = y'
| r = \cos x
| c = Derivative of Sine Function
}}
{{eqn | ll= \leadsto
| l = y' '
| r = -\sin x
| c = Derivative of Cosine Function
}}
{{end-eqn}}
and so:
:$y_1 = x$
is a particu... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' + y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 \sin x + C_2 \cos x$ | Note that:
{{begin-eqn}}
{{eqn | l = y_1
| r = \sin x
| c =
}}
{{eqn | ll= \leadsto
| l = y'
| r = \cos x
| c = [[Derivative of Sine Function]]
}}
{{eqn | ll= \leadsto
| l = y' '
| r = -\sin x
| c = [[Derivative of Cosine Function]]
}}
{{end-eqn}}
and so:
:$y_1 = x$
is... | Linear Second Order ODE/y'' + y = 0/Proof 4 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_y_=_0/Proof_4 | [
"Linear Second Order ODE/y'' + y = 0",
"Examples of Linear Second Order ODE/y'' + k^2 y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Derivative of Sine Function",
"Derivative of Cosine Function",
"Definition:Differential Equation/Solution/Particular Solution",
"Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Primitive (Calcu... |
proofwiki-11119 | Linear Second Order ODE/y'' + y = 0 | The second order ODE:
:$(1): \quad y' ' + y = 0$
has the general solution:
:$y = C_1 \sin x + C_2 \cos x$ | Taking Laplace transforms, we have:
:$\laptrans {y' ' + y} = \laptrans 0$
From Laplace Transform of Constant Mapping, we have:
:$\laptrans 0 = 0$
We also have:
{{begin-eqn}}
{{eqn | l = \laptrans {y' ' + y}
| r = \laptrans {y' '} + \laptrans y
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r =... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' + y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 \sin x + C_2 \cos x$ | Taking [[Definition:Laplace Transform|Laplace transforms]], we have:
:$\laptrans {y' ' + y} = \laptrans 0$
From [[Laplace Transform of Constant Mapping]], we have:
:$\laptrans 0 = 0$
We also have:
{{begin-eqn}}
{{eqn | l = \laptrans {y' ' + y}
| r = \laptrans {y' '} + \laptrans y
| c = [[Linear Comb... | Linear Second Order ODE/y'' + y = 0/Proof 5 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_y_=_0/Proof_5 | [
"Linear Second Order ODE/y'' + y = 0",
"Examples of Linear Second Order ODE/y'' + k^2 y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Laplace Transform",
"Laplace Transform of Constant Mapping",
"Linear Combination of Laplace Transforms",
"Laplace Transform of Second Derivative",
"Linear Combination of Laplace Transforms",
"Laplace Transform of Cosine",
"Laplace Transform of Sine"
] |
proofwiki-11120 | Linear Second Order ODE/y'' + y = 0/y(0) = 2, y'(0) = 3 | The second order ODE:
:$(1): \quad y' ' + y = 0$
with initial conditions:
:$\map y 0 = 2$
:$\map {y'} 0 = 3$
has the particular solution:
:$y = 3 \sin x + 2 \cos x$ | From Linear Second Order ODE: $y' ' + y = 0$, the general solution of $(1)$ is:
:$y = C_1 \sin x + C_2 \cos x$
Differentiating {{WRT|Differentiation}} $x$:
:$y' = C_1 \cos x - C_2 \sin x$
Thus for the initial conditions:
{{begin-eqn}}
{{eqn | l = \map y 0
| r = C_1 \sin 0 + C_2 \cos 0
| c =
}}
{{eqn | r = ... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' + y = 0$
with [[Definition:Initial Condition|initial conditions]]:
:$\map y 0 = 2$
:$\map {y'} 0 = 3$
has the [[Definition:Particular Solution of Differential Equation|particular solution]]:
:$y = 3 \sin x + 2 \cos x$ | From [[Linear Second Order ODE/y'' + y = 0|Linear Second Order ODE: $y' ' + y = 0$]], the [[Definition:General Solution to Differential Equation|general solution]] of $(1)$ is:
:$y = C_1 \sin x + C_2 \cos x$
[[Definition:Differentiation|Differentiating]] {{WRT|Differentiation}} $x$:
:$y' = C_1 \cos x - C_2 \sin x$
Th... | Linear Second Order ODE/y'' + y = 0/y(0) = 2, y'(0) = 3 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_y_=_0/y(0)_=_2,_y'(0)_=_3 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_y_=_0/y(0)_=_2,_y'(0)_=_3 | [
"Linear Second Order ODE/y'' + y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Initial Condition",
"Definition:Differential Equation/Solution/Particular Solution"
] | [
"Linear Second Order ODE/y'' + y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Differentiation",
"Definition:Initial Condition",
"Sine of Zero is Zero",
"Cosine of Zero is One",
"Sine of Zero is Zero",
"Cosine of Zero is One"
] |
proofwiki-11121 | Linearly Independent Solutions of y'' - y = 0 | The second order ODE:
:$(1): \quad y' ' - y = 0$
has solutions:
:$y_1 = e^x$
:$y_2 = e^{-x}$
which are linearly independent. | We have that:
{{begin-eqn}}
{{eqn | l = \frac \d {\d x} \, e^x
| r = e^x
| c = Derivative of Exponential Function
}}
{{eqn | l = \frac \d {\d x} \, e^{-x}
| r = -e^{-x}
| c = Derivative of Exponential Function
}}
{{eqn | ll= \leadsto
| l = \frac {\d^2} {\d x^2} \, e^x
| r = e^x
... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' - y = 0$
has [[Definition:Solution to Differential Equation|solutions]]:
:$y_1 = e^x$
:$y_2 = e^{-x}$
which are [[Definition:Linearly Independent Real Functions|linearly independent]]. | We have that:
{{begin-eqn}}
{{eqn | l = \frac \d {\d x} \, e^x
| r = e^x
| c = [[Derivative of Exponential Function]]
}}
{{eqn | l = \frac \d {\d x} \, e^{-x}
| r = -e^{-x}
| c = [[Derivative of Exponential Function]]
}}
{{eqn | ll= \leadsto
| l = \frac {\d^2} {\d x^2} \, e^x
| r = e... | Linearly Independent Solutions of y'' - y = 0 | https://proofwiki.org/wiki/Linearly_Independent_Solutions_of_y''_-_y_=_0 | https://proofwiki.org/wiki/Linearly_Independent_Solutions_of_y''_-_y_=_0 | [
"Linear Second Order ODE/y'' - y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution",
"Definition:Linearly Independent Real Functions"
] | [
"Derivative of Exponential Function",
"Derivative of Exponential Function",
"Definition:Differential Equation/Solution",
"Definition:Wronskian",
"Definition:Wronskian",
"Zero Wronskian of Solutions of Homogeneous Linear Second Order ODE iff Linearly Dependent",
"Definition:Linearly Independent Real Func... |
proofwiki-11122 | Linear Second Order ODE/x^2 y'' - 2 x y' + 2 y = 0 | The second order ODE:
:$(1): \quad x^2 y' ' - 2 x y' + 2 y = 0$
has the general solution:
:$y = C_1 x + C_2 x^2$
on any closed real interval which does not contain $0$. | Consider the functions:
:$\map {y_1} x = x$
:$\map {y_2} x = x^2$
We have that:
{{begin-eqn}}
{{eqn | l = \frac \d {\d x} \, x
| r = 1
| c = Power Rule for Derivatives
}}
{{eqn | l = \frac \d {\d x} \, x^2
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \frac {\d^2... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad x^2 y' ' - 2 x y' + 2 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 x + C_2 x^2$
on any [[Definition:Closed Real Interval|closed real interval]] which does not contain $0$. | Consider the [[Definition:Real Function|functions]]:
:$\map {y_1} x = x$
:$\map {y_2} x = x^2$
We have that:
{{begin-eqn}}
{{eqn | l = \frac \d {\d x} \, x
| r = 1
| c = [[Power Rule for Derivatives]]
}}
{{eqn | l = \frac \d {\d x} \, x^2
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eq... | Linear Second Order ODE/x^2 y'' - 2 x y' + 2 y = 0/Proof 1 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/x^2_y''_-_2_x_y'_+_2_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/x^2_y''_-_2_x_y'_+_2_y_=_0/Proof_1 | [
"Examples of Cauchy-Euler Equation",
"Linear Second Order ODE/x^2 y'' - 2 x y' + 2 y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Real Interval/Closed"
] | [
"Definition:Real Function",
"Power Rule for Derivatives",
"Power Rule for Derivatives",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Wronskian",
"Definition:Wronskian",
"Definition:Real Interval/Closed",
"Zero Wronskian of Solutions of Homogeneous Linear Second Order OD... |
proofwiki-11123 | Linear Second Order ODE/x^2 y'' - 2 x y' + 2 y = 0 | The second order ODE:
:$(1): \quad x^2 y' ' - 2 x y' + 2 y = 0$
has the general solution:
:$y = C_1 x + C_2 x^2$
on any closed real interval which does not contain $0$. | It can be seen that $(1)$ is an instance of the Cauchy-Euler Equation:
:$x^2 y' ' + p x y' + q y = 0$
where:
:$p = -2$
:$q = 2$
By Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE, this can be expressed as:
:$\dfrac {\d^2 y} {\d t^2} + \paren {p - 1} \dfrac {\d y} {\d t} + q y = 0$
by making the s... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad x^2 y' ' - 2 x y' + 2 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 x + C_2 x^2$
on any [[Definition:Closed Real Interval|closed real interval]] which does not contain $0$. | It can be seen that $(1)$ is an instance of the [[Definition:Cauchy-Euler Equation|Cauchy-Euler Equation]]:
:$x^2 y' ' + p x y' + q y = 0$
where:
:$p = -2$
:$q = 2$
By [[Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE]], this can be expressed as:
:$\dfrac {\d^2 y} {\d t^2} + \paren {p - 1} \dfr... | Linear Second Order ODE/x^2 y'' - 2 x y' + 2 y = 0/Proof 2 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/x^2_y''_-_2_x_y'_+_2_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/x^2_y''_-_2_x_y'_+_2_y_=_0/Proof_2 | [
"Examples of Cauchy-Euler Equation",
"Linear Second Order ODE/x^2 y'' - 2 x y' + 2 y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Real Interval/Closed"
] | [
"Definition:Cauchy-Euler Equation",
"Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE",
"Linear Second Order ODE/y'' - 3 y' + 2 y = 0",
"Definition:Differential Equation/Solution/General Solution"
] |
proofwiki-11124 | Linear Second Order ODE/x^2 y'' - 2 x y' + 2 y = 0/y(1) = 3, y'(1) = 5 | The second order ODE:
:$(1): \quad x^2 y'' - 2 x y' + 2 y = 0$
with initial conditions:
{{begin-eqn}}
{{eqn | l = \map y 1
| r = 3
}}
{{eqn | l = \map {y'} 1
| r = 5
}}
{{end-eqn}}
has the particular solution:
:$y = x + 2 x^2$ | From Linear Second Order ODE: $x^2 y'' - 2 x y' + 2 y = 0$, the general solution of $(1)$ is:
:$y = C_1 x + C_2 x^2$
Differentiating {{WRT|Differentiation}} $x$:
:$y' = C_1 + 2 C_2 x$
Thus for the initial conditions:
{{begin-eqn}}
{{eqn | l = \map y 1
| r = C_1 \times 1 + C_2 \times 1^2
| c =
}}
{{eqn | r ... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad x^2 y'' - 2 x y' + 2 y = 0$
with [[Definition:Initial Condition|initial conditions]]:
{{begin-eqn}}
{{eqn | l = \map y 1
| r = 3
}}
{{eqn | l = \map {y'} 1
| r = 5
}}
{{end-eqn}}
has the [[Definition:Particular Solution of Differential Equa... | From [[Linear Second Order ODE/x^2 y'' - 2 x y' + 2 y = 0|Linear Second Order ODE: $x^2 y'' - 2 x y' + 2 y = 0$]], the [[Definition:General Solution to Differential Equation|general solution]] of $(1)$ is:
:$y = C_1 x + C_2 x^2$
[[Definition:Differentiation|Differentiating]] {{WRT|Differentiation}} $x$:
:$y' = C_1 + 2 ... | Linear Second Order ODE/x^2 y'' - 2 x y' + 2 y = 0/y(1) = 3, y'(1) = 5 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/x^2_y''_-_2_x_y'_+_2_y_=_0/y(1)_=_3,_y'(1)_=_5 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/x^2_y''_-_2_x_y'_+_2_y_=_0/y(1)_=_3,_y'(1)_=_5 | [
"Linear Second Order ODE/x^2 y'' - 2 x y' + 2 y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Initial Condition",
"Definition:Differential Equation/Solution/Particular Solution"
] | [
"Linear Second Order ODE/x^2 y'' - 2 x y' + 2 y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Differentiation",
"Definition:Initial Condition"
] |
proofwiki-11125 | Linear Second Order ODE/y'' - 3 y' + 2 y = 0 | The second order ODE:
:$(1): \quad y' ' - 3 y' + 2 y = 0$
has the general solution:
:$y = C_1 e^x + C_2 e^{2 x}$ | Consider the functions:
:$\map {y_1} x = e^x$
:$\map {y_2} x = e^{2 x}$
We have that:
{{begin-eqn}}
{{eqn | l = \frac {\d} {\d x} \, e^x
| r = e^x
| c = Power Rule for Derivatives
}}
{{eqn | l = \frac {\d} {\d x} \, e^{2 x}
| r = 2 e^{2 x}
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' - 3 y' + 2 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^x + C_2 e^{2 x}$ | Consider the [[Definition:Real Function|functions]]:
:$\map {y_1} x = e^x$
:$\map {y_2} x = e^{2 x}$
We have that:
{{begin-eqn}}
{{eqn | l = \frac {\d} {\d x} \, e^x
| r = e^x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | l = \frac {\d} {\d x} \, e^{2 x}
| r = 2 e^{2 x}
| c = [[Power Rule f... | Linear Second Order ODE/y'' - 3 y' + 2 y = 0/Proof 1 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_3_y'_+_2_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_3_y'_+_2_y_=_0/Proof_1 | [
"Linear Second Order ODE/y'' - 3 y' + 2 y = 0",
"Examples of Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Real Function",
"Power Rule for Derivatives",
"Power Rule for Derivatives",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Wronskian",
"Definition:Wronskian",
"Zero Wronskian of Solutions of Homogeneous Linear Second Order ODE iff Linearly Dependent",
"Definit... |
proofwiki-11126 | Linear Second Order ODE/y'' - 3 y' + 2 y = 0 | The second order ODE:
:$(1): \quad y' ' - 3 y' + 2 y = 0$
has the general solution:
:$y = C_1 e^x + C_2 e^{2 x}$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad: m^2 - 3 m + 2 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = 1$
:$m_2 = 2$
These are real and unequal.
So from Solution of Constant Coeffici... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' - 3 y' + 2 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^x + C_2 e^{2 x}$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad: m^2 - 3 m + 2 = 0$
From [[Solution to Quadratic Equation with Real Coefficients]]... | Linear Second Order ODE/y'' - 3 y' + 2 y = 0/Proof 2 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_3_y'_+_2_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_3_y'_+_2_y_=_0/Proof_2 | [
"Linear Second Order ODE/y'' - 3 y' + 2 y = 0",
"Examples of Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Definition:Real Number",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential Equ... |
proofwiki-11127 | Linear Second Order ODE/y'' - 3 y' + 2 y = 0/y(0) = -1, y'(0) = 1 | The second order ODE:
:$(1): \quad y' ' - 3 y' + 2 y = 0$
with initial conditions:
:$\map y 0 = -1$
:$\map {y'} 0 = 1$
has the particular solution:
:$y = -3 e^x + 2 e^{2 x}$ | From Linear Second Order ODE: $y' ' - 3 y' + 2 y = 0$, the general solution of $(1)$ is:
:$y = C_1 e^x + C_2 e^{2 x}$
Differentiating {{WRT|Differentiation}} $x$:
:$y' = C_1 e^x + 2 C_2 e^{2 x}$
Thus for the initial conditions:
{{begin-eqn}}
{{eqn | l = \map y 0
| r = C_1 e^0 + C_2 e^{2 \times 0}
| c =
}}
... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' - 3 y' + 2 y = 0$
with [[Definition:Initial Condition|initial conditions]]:
:$\map y 0 = -1$
:$\map {y'} 0 = 1$
has the [[Definition:Particular Solution of Differential Equation|particular solution]]:
:$y = -3 e^x + 2 e^{2 x}$ | From [[Linear Second Order ODE/y'' - 3 y' + 2 y = 0|Linear Second Order ODE: $y' ' - 3 y' + 2 y = 0$]], the [[Definition:General Solution to Differential Equation|general solution]] of $(1)$ is:
:$y = C_1 e^x + C_2 e^{2 x}$
[[Definition:Differentiation|Differentiating]] {{WRT|Differentiation}} $x$:
:$y' = C_1 e^x + 2 C... | Linear Second Order ODE/y'' - 3 y' + 2 y = 0/y(0) = -1, y'(0) = 1 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_3_y'_+_2_y_=_0/y(0)_=_-1,_y'(0)_=_1 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_3_y'_+_2_y_=_0/y(0)_=_-1,_y'(0)_=_1 | [
"Linear Second Order ODE/y'' - 3 y' + 2 y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Initial Condition",
"Definition:Differential Equation/Solution/Particular Solution"
] | [
"Linear Second Order ODE/y'' - 3 y' + 2 y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Differentiation",
"Definition:Initial Condition"
] |
proofwiki-11128 | Linear Second Order ODE/y'' - 4 y' + 4 y = 0 | The second order ODE:
:$(1): \quad y' ' - 4 y' + 4 y = 0$
has the general solution:
:$y = C_1 e^{2 x} + C_2 x e^{2 x}$ | Consider the functions:
:$\map {y_1} x = e^{2 x}$
:$\map {y_2} x = x e^{2 x}$
We have that:
{{begin-eqn}}
{{eqn | l = \frac \d {\d x} \, e^{2 x}
| r = 2 e^{2 x}
| c = Power Rule for Derivatives
}}
{{eqn | l = \frac \d {\d x} \, x e^{2 x}
| r = 2 x e^{2 x} + e^{2 x}
| c = Power Rule for Derivativ... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' - 4 y' + 4 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^{2 x} + C_2 x e^{2 x}$ | Consider the [[Definition:Real Function|functions]]:
:$\map {y_1} x = e^{2 x}$
:$\map {y_2} x = x e^{2 x}$
We have that:
{{begin-eqn}}
{{eqn | l = \frac \d {\d x} \, e^{2 x}
| r = 2 e^{2 x}
| c = [[Power Rule for Derivatives]]
}}
{{eqn | l = \frac \d {\d x} \, x e^{2 x}
| r = 2 x e^{2 x} + e^{2 x}
... | Linear Second Order ODE/y'' - 4 y' + 4 y = 0/Proof 1 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_4_y'_+_4_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_4_y'_+_4_y_=_0/Proof_1 | [
"Examples of Constant Coefficient Homogeneous LSOODEs",
"Linear Second Order ODE/y'' - 4 y' + 4 y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Real Function",
"Power Rule for Derivatives",
"Power Rule for Derivatives",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Wronskian",
"Definition:Wronskian",
"Zero Wronskian of Solutions of Homogeneous Linear Second Order ODE iff Linearly Dependent",
"Definit... |
proofwiki-11129 | Linear Second Order ODE/y'' - 4 y' + 4 y = 0 | The second order ODE:
:$(1): \quad y' ' - 4 y' + 4 y = 0$
has the general solution:
:$y = C_1 e^{2 x} + C_2 x e^{2 x}$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad: m^2 - 4 m + 4 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = m_2 = 2$
These are real and equal.
So from Solution of Constant Coefficient Hom... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' - 4 y' + 4 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^{2 x} + C_2 x e^{2 x}$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad: m^2 - 4 m + 4 = 0$
From [[Solution to Quadratic Equation with Real Coefficients]],... | Linear Second Order ODE/y'' - 4 y' + 4 y = 0/Proof 2 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_4_y'_+_4_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_4_y'_+_4_y_=_0/Proof_2 | [
"Examples of Constant Coefficient Homogeneous LSOODEs",
"Linear Second Order ODE/y'' - 4 y' + 4 y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Definition:Real Number",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential Equ... |
proofwiki-11130 | Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another | Let $\map {y_1} x$ be a particular solution to the homogeneous linear second order ODE:
:$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$
such that $y_1$ is not the trivial solution.
Then there exists a standard procedure to determine another particular solution $\map {y_2} x$ of $... | Let $\map {y_1} x$ be a non-trivial particular solution to $(1)$.
Thus, for all $C \in \R$, $C y_1$ is also a non-trivial particular solution to $(1)$.
Let $\map v x$ be a function of $x$ such that:
:$(2): \quad \map {y_2} x = \map v x \, \map {y_1} x$
is a particular solution to $(1)$ such that $y_1$ and $y_2$ are lin... | Let $\map {y_1} x$ be a [[Definition:Particular Solution|particular solution]] to the [[Definition:Homogeneous Linear Second Order ODE|homogeneous linear second order ODE]]:
:$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$
such that $y_1$ is not the [[Trivial Solution of Homogeneo... | Let $\map {y_1} x$ be a [[Trivial Solution of Homogeneous Linear 2nd Order ODE|non-trivial]] [[Definition:Particular Solution|particular solution]] to $(1)$.
Thus, for all $C \in \R$, $C y_1$ is also a [[Trivial Solution of Homogeneous Linear 2nd Order ODE|non-trivial]] [[Definition:Particular Solution|particular solu... | Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another | https://proofwiki.org/wiki/Particular_Solution_to_Homogeneous_Linear_Second_Order_ODE_gives_rise_to_Another | https://proofwiki.org/wiki/Particular_Solution_to_Homogeneous_Linear_Second_Order_ODE_gives_rise_to_Another | [
"Homogeneous LSOODEs"
] | [
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Homogeneous Linear Second Order ODE",
"Trivial Solution of Homogeneous Linear 2nd Order ODE",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Linearly Independent Real Functions"
] | [
"Trivial Solution of Homogeneous Linear 2nd Order ODE",
"Definition:Differential Equation/Solution/Particular Solution",
"Trivial Solution of Homogeneous Linear 2nd Order ODE",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Real Function",
"Definition:Differential Equation/So... |
proofwiki-11131 | Induced Solution to Homogeneous Linear Second Order ODE is Linearly Independent with Inducing Solution | Let $\map {y_1} x$ be a particular solution to the homogeneous linear second order ODE:
:$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$
such that $y_1$ is not the trivial solution.
Let $\map {y_2} x$ be the real function defined as:
:$\map {y_2} x = \map v x \map {y_1} x$
where:
... | This will be demonstrated by calculating the Wronskian of $y_1$ and $y_2$ and demonstrating that it is non-zero everywhere.
First we take the derivative of $v$ {{WRT|Differentiation}} $x$:
:$v' = \dfrac 1 { {y_1}^2} e^{- \int P \rd x}$
{{begin-eqn}}
{{eqn | l = \map W {y_1, y_2}
| r = y_1 {y_2}' - y_2 {y_1}'
... | Let $\map {y_1} x$ be a [[Definition:Particular Solution|particular solution]] to the [[Definition:Homogeneous Linear Second Order ODE|homogeneous linear second order ODE]]:
:$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$
such that $y_1$ is not the [[Trivial Solution of Homogeneo... | This will be demonstrated by calculating the [[Definition:Wronskian|Wronskian]] of $y_1$ and $y_2$ and demonstrating that it is non-zero everywhere.
First we take the [[Definition:Derivative|derivative]] of $v$ {{WRT|Differentiation}} $x$:
:$v' = \dfrac 1 { {y_1}^2} e^{- \int P \rd x}$
{{begin-eqn}}
{{eqn | l = \map... | Induced Solution to Homogeneous Linear Second Order ODE is Linearly Independent with Inducing Solution | https://proofwiki.org/wiki/Induced_Solution_to_Homogeneous_Linear_Second_Order_ODE_is_Linearly_Independent_with_Inducing_Solution | https://proofwiki.org/wiki/Induced_Solution_to_Homogeneous_Linear_Second_Order_ODE_is_Linearly_Independent_with_Inducing_Solution | [
"Homogeneous LSOODEs"
] | [
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Homogeneous Linear Second Order ODE",
"Trivial Solution of Homogeneous Linear 2nd Order ODE",
"Definition:Real Function",
"Definition:Linearly Independent Real Functions"
] | [
"Definition:Wronskian",
"Definition:Derivative",
"Product Rule for Derivatives",
"Definition:Real Function",
"Zero Wronskian of Solutions of Homogeneous Linear Second Order ODE iff Linearly Dependent",
"Definition:Linearly Independent Real Functions"
] |
proofwiki-11132 | Linear Second Order ODE/x^2 y'' + x y' - y = 0 | The second order ODE:
:$(1): \quad x^2 y'' + x y' - y = 0$
has the general solution:
:$y = C_1 x + \dfrac {C_2} x$ | The particular solution:
:$y_1 = x$
can be found by inspection.
Let $(1)$ be written as:
:$(2): \quad y'' + \dfrac {y'} x - \dfrac y {x^2} = 0$
which is in the form:
:$y'' + \map P x y' + \map Q x y = 0$
where:
:$\map P x = \dfrac 1 x$
:$\map Q x = \dfrac 1 {x^2}$
From Particular Solution to Homogeneous Linear Second O... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad x^2 y'' + x y' - y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 x + \dfrac {C_2} x$ | The [[Definition:Particular Solution|particular solution]]:
:$y_1 = x$
can be found by inspection.
Let $(1)$ be written as:
:$(2): \quad y'' + \dfrac {y'} x - \dfrac y {x^2} = 0$
which is in the form:
:$y'' + \map P x y' + \map Q x y = 0$
where:
:$\map P x = \dfrac 1 x$
:$\map Q x = \dfrac 1 {x^2}$
From [[Particula... | Linear Second Order ODE/x^2 y'' + x y' - y = 0/Proof 1 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/x^2_y''_+_x_y'_-_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/x^2_y''_+_x_y'_-_y_=_0/Proof_1 | [
"Examples of Cauchy-Euler Equation",
"Linear Second Order ODE/x^2 y'' + x y' - y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Differential Equation/Solution/Particular Solution",
"Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another",
"Definition:Differential Equation/Solution/Particular Solution",
"Two Linearly Independent Solutions of Homogeneous Linear Second Order ODE generate General Solu... |
proofwiki-11133 | Linear Second Order ODE/x^2 y'' + x y' - y = 0 | The second order ODE:
:$(1): \quad x^2 y'' + x y' - y = 0$
has the general solution:
:$y = C_1 x + \dfrac {C_2} x$ | It can be seen that $(1)$ is an instance of the Cauchy-Euler Equation:
:$x^2 y'' + p x y' + q y = 0$
where:
:$p = 1$
:$q = -1$
By Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE, this can be expressed as:
:$\dfrac {\d^2 y} {\d t^2} + \paren {p - 1} \dfrac {\d y} {\d t^2} + q y = 0$
by making the ... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad x^2 y'' + x y' - y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 x + \dfrac {C_2} x$ | It can be seen that $(1)$ is an instance of the [[Definition:Cauchy-Euler Equation|Cauchy-Euler Equation]]:
:$x^2 y'' + p x y' + q y = 0$
where:
:$p = 1$
:$q = -1$
By [[Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE]], this can be expressed as:
:$\dfrac {\d^2 y} {\d t^2} + \paren {p - 1} \dfra... | Linear Second Order ODE/x^2 y'' + x y' - y = 0/Proof 2 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/x^2_y''_+_x_y'_-_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/x^2_y''_+_x_y'_-_y_=_0/Proof_2 | [
"Examples of Cauchy-Euler Equation",
"Linear Second Order ODE/x^2 y'' + x y' - y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Cauchy-Euler Equation",
"Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE",
"Linear Second Order ODE/y'' - y = 0",
"Definition:Differential Equation/Solution/General Solution"
] |
proofwiki-11134 | Linear Second Order ODE/y'' - y = 0 | The second order ODE:
:$(1): \quad y' ' - y = 0$
has the general solution:
:$y = C_1 e^x + C_2 e^{-x}$ | Note that:
{{begin-eqn}}
{{eqn | l = y_1
| r = e^x
| c =
}}
{{eqn | ll= \leadsto
| l = y'
| r = e^x
| c = Derivative of Exponential Function
}}
{{eqn | ll= \leadsto
| l = y' '
| r = e^x
| c = Derivative of Exponential Function
}}
{{end-eqn}}
and so by inspection:
:$y_1 =... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' - y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^x + C_2 e^{-x}$ | Note that:
{{begin-eqn}}
{{eqn | l = y_1
| r = e^x
| c =
}}
{{eqn | ll= \leadsto
| l = y'
| r = e^x
| c = [[Derivative of Exponential Function]]
}}
{{eqn | ll= \leadsto
| l = y' '
| r = e^x
| c = [[Derivative of Exponential Function]]
}}
{{end-eqn}}
and so by inspectio... | Linear Second Order ODE/y'' - y = 0/Proof 1 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_y_=_0/Proof_1 | [
"Linear Second Order ODE/y'' - y = 0",
"Examples of Linear Second Order ODE/y'' - k^2 y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Derivative of Exponential Function",
"Derivative of Exponential Function",
"Definition:Differential Equation/Solution/Particular Solution",
"Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Prim... |
proofwiki-11135 | Linear Second Order ODE/y'' - y = 0 | The second order ODE:
:$(1): \quad y' ' - y = 0$
has the general solution:
:$y = C_1 e^x + C_2 e^{-x}$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad: m^2 - 1 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = 1$
:$m_2 = -1$
These are real and unequal.
So from Solution of Constant Coefficient H... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' - y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^x + C_2 e^{-x}$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad: m^2 - 1 = 0$
From [[Solution to Quadratic Equation with Real Coefficients]], the [... | Linear Second Order ODE/y'' - y = 0/Proof 2 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_y_=_0/Proof_2 | [
"Linear Second Order ODE/y'' - y = 0",
"Examples of Linear Second Order ODE/y'' - k^2 y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Definition:Real Number",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential Equ... |
proofwiki-11136 | Linear Second Order ODE/y'' - y = 0 | The second order ODE:
:$(1): \quad y' ' - y = 0$
has the general solution:
:$y = C_1 e^x + C_2 e^{-x}$ | This is an instance of:
:Linear Second Order ODE: $y' ' - k^2 y = 0$
which yields:
:$y = C_1 e^{k x} + C_2 e^{-k x}$
where $k = 1$.
Hence the result.
{{qed}} | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' - y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^x + C_2 e^{-x}$ | This is an instance of:
:[[Linear Second Order ODE/y'' - k^2 y = 0|Linear Second Order ODE: $y' ' - k^2 y = 0$]]
which yields:
:$y = C_1 e^{k x} + C_2 e^{-k x}$
where $k = 1$.
Hence the result.
{{qed}} | Linear Second Order ODE/y'' - y = 0/Proof 3 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_y_=_0/Proof_3 | [
"Linear Second Order ODE/y'' - y = 0",
"Examples of Linear Second Order ODE/y'' - k^2 y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Linear Second Order ODE/y'' - k^2 y = 0"
] |
proofwiki-11137 | Second Order ODE/x y'' + 3 y' = 0 | The second order ODE:
:$(1): \quad x y' ' + 3 y' = 0$
has the general solution:
:$y = C_1 + \dfrac {C_2} {x^2}$ | Note that:
{{begin-eqn}}
{{eqn | l = y_1
| r = 1
| c =
}}
{{eqn | ll= \leadsto
| l = y'
| r = 0
| c = Derivative of Constant
}}
{{eqn | ll= \leadsto
| l = y' '
| r = 0
| c = Derivative of Constant
}}
{{end-eqn}}
and so by inspection:
:$y_1 = 1$
is a particular solution o... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad x y' ' + 3 y' = 0$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$y = C_1 + \dfrac {C_2} {x^2}$ | Note that:
{{begin-eqn}}
{{eqn | l = y_1
| r = 1
| c =
}}
{{eqn | ll= \leadsto
| l = y'
| r = 0
| c = [[Derivative of Constant]]
}}
{{eqn | ll= \leadsto
| l = y' '
| r = 0
| c = [[Derivative of Constant]]
}}
{{end-eqn}}
and so by inspection:
:$y_1 = 1$
is a [[Definitio... | Second Order ODE/x y'' + 3 y' = 0 | https://proofwiki.org/wiki/Second_Order_ODE/x_y''_+_3_y'_=_0 | https://proofwiki.org/wiki/Second_Order_ODE/x_y''_+_3_y'_=_0 | [
"Examples of Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Derivative of Constant",
"Derivative of Constant",
"Definition:Differential Equation/Solution/Particular Solution",
"Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another",
"Definition:Differential Equation/Solution/Particular Solution",
"Primitive of Reciprocal",
"Two Linear... |
proofwiki-11138 | Linear Second Order ODE/x^2 y'' + x y' - 4 y = 0 | The second order ODE:
:$(1): \quad x^2 y'' + x y' - 4 y = 0$
has the general solution:
:$y = C_1 x^2 + \dfrac {C_2} {x^2}$ | Note that:
{{begin-eqn}}
{{eqn | l = y_1
| r = x^2
| c =
}}
{{eqn | ll= \leadsto
| l = y'
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = y''
| r = 2
| c = Power Rule for Derivatives
}}
{{end-eqn}}
and so by inspection:
:$y_1 = x^2$
is a particul... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad x^2 y'' + x y' - 4 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 x^2 + \dfrac {C_2} {x^2}$ | Note that:
{{begin-eqn}}
{{eqn | l = y_1
| r = x^2
| c =
}}
{{eqn | ll= \leadsto
| l = y'
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = y''
| r = 2
| c = [[Power Rule for Derivatives]]
}}
{{end-eqn}}
and so by inspection:
:$y_1 = x^2$
is ... | Linear Second Order ODE/x^2 y'' + x y' - 4 y = 0/Proof 1 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/x^2_y''_+_x_y'_-_4_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/x^2_y''_+_x_y'_-_4_y_=_0/Proof_1 | [
"Examples of Cauchy-Euler Equation",
"Linear Second Order ODE/x^2 y'' + x y' - 4 y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Power Rule for Derivatives",
"Power Rule for Derivatives",
"Definition:Differential Equation/Solution/Particular Solution",
"Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another",
"Definition:Differential Equation/Solution/Particular Solution",
"Primitive of Reciprocal",
"Tw... |
proofwiki-11139 | Linear Second Order ODE/x^2 y'' + x y' - 4 y = 0 | The second order ODE:
:$(1): \quad x^2 y'' + x y' - 4 y = 0$
has the general solution:
:$y = C_1 x^2 + \dfrac {C_2} {x^2}$ | It can be seen that $(1)$ is an instance of the Cauchy-Euler Equation:
:$x^2 y'' + p x y' + q y = 0$
where:
:$p = 1$
:$q = -4$
By Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE, this can be expressed as:
:$\dfrac {\d^2 y} {\d t^2} + \paren {p - 1} \dfrac {\d y} {\d t^2} + q y = 0$
by making the ... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad x^2 y'' + x y' - 4 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 x^2 + \dfrac {C_2} {x^2}$ | It can be seen that $(1)$ is an instance of the [[Definition:Cauchy-Euler Equation|Cauchy-Euler Equation]]:
:$x^2 y'' + p x y' + q y = 0$
where:
:$p = 1$
:$q = -4$
By [[Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE]], this can be expressed as:
:$\dfrac {\d^2 y} {\d t^2} + \paren {p - 1} \dfra... | Linear Second Order ODE/x^2 y'' + x y' - 4 y = 0/Proof 2 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/x^2_y''_+_x_y'_-_4_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/x^2_y''_+_x_y'_-_4_y_=_0/Proof_2 | [
"Examples of Cauchy-Euler Equation",
"Linear Second Order ODE/x^2 y'' + x y' - 4 y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Cauchy-Euler Equation",
"Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE",
"Linear Second Order ODE/y'' - 4 y = 0",
"Definition:Differential Equation/Solution/General Solution"
] |
proofwiki-11140 | Legendre's Differential Equation/(1 - x^2) y'' - 2 x y' + 2 y = 0 | The special case of Legendre's differential equation:
:$(1): \quad \paren {1 - x^2} y' ' - 2 x y' + 2 y = 0$
has the general solution:
:$y = C_1 x + C_2 \paren {\dfrac x 2 \, \map \ln {\dfrac {1 + x} {1 - x} } - 1}$ | Note that:
{{begin-eqn}}
{{eqn | l = y_1
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = {y_1}'
| r = 1
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = {y_1}' '
| r = 0
| c = Derivative of Constant
}}
{{end-eqn}}
and so by inspection:
:$y_1 = x$
is a particula... | The special case of [[Definition:Legendre's Differential Equation|Legendre's differential equation]]:
:$(1): \quad \paren {1 - x^2} y' ' - 2 x y' + 2 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 x + C_2 \paren {\dfrac x 2 \, \map \ln {\dfrac {1 + x} {1 - x} } - 1}... | Note that:
{{begin-eqn}}
{{eqn | l = y_1
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = {y_1}'
| r = 1
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = {y_1}' '
| r = 0
| c = [[Derivative of Constant]]
}}
{{end-eqn}}
and so by inspection:
:$y_1 = x$
is a... | Legendre's Differential Equation/(1 - x^2) y'' - 2 x y' + 2 y = 0 | https://proofwiki.org/wiki/Legendre's_Differential_Equation/(1_-_x^2)_y''_-_2_x_y'_+_2_y_=_0 | https://proofwiki.org/wiki/Legendre's_Differential_Equation/(1_-_x^2)_y''_-_2_x_y'_+_2_y_=_0 | [
"Examples of Legendre's Differential Equation"
] | [
"Definition:Legendre's Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Power Rule for Derivatives",
"Derivative of Constant",
"Definition:Differential Equation/Solution/Particular Solution",
"Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another",
"Definition:Differential Equation/Solution/Particular Solution",
"Primitive of Function under its Der... |
proofwiki-11141 | Topological Subspace of Real Number Line is Lindelöf | Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Let $W$ be a non-empty subset of $\R$.
Then $R_W$ is Lindelöf
where $R_W$ denotes the topological subspace of $R$ on $W$. | Let $\CC$ be a open cover for $W$.
Define $Q := \set {\openint a b: a, b \in \Q}$
Define a mapping $h: Q \to \Q \times \Q$:
:$\forall \openint a b \in Q: \map h {\openint a b} = \tuple {a, b}$
It is easy to see by definition that
:$h$ is an injection.
By Injection iff Cardinal Inequality:
:$\card Q \le \card {\Q \times... | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Let $W$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $\R$.
Then $R_W$ is [[Definition:Lindelöf Space|Lindelöf]]
where $R_W$ denotes the [[Defi... | Let $\CC$ be a [[Definition:Open Cover|open cover]] for $W$.
Define $Q := \set {\openint a b: a, b \in \Q}$
Define a [[Definition:Mapping|mapping]] $h: Q \to \Q \times \Q$:
:$\forall \openint a b \in Q: \map h {\openint a b} = \tuple {a, b}$
It is easy to see by definition that
:$h$ is an [[Definition:Injection|inje... | Topological Subspace of Real Number Line is Lindelöf | https://proofwiki.org/wiki/Topological_Subspace_of_Real_Number_Line_is_Lindelöf | https://proofwiki.org/wiki/Topological_Subspace_of_Real_Number_Line_is_Lindelöf | [
"Lindelöf Spaces",
"Real Number Line with Euclidean Topology"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Lindelöf Space",
"Definition:Topological Subspace"
] | [
"Definition:Open Cover",
"Definition:Mapping",
"Definition:Injection",
"Injection iff Cardinal Inequality",
"Definition:Cardinality",
"Rational Numbers are Countably Infinite",
"Definition:Countably Infinite/Set",
"Definition:Countably Infinite/Set",
"Definition:Bijection",
"Definition:Set Equalit... |
proofwiki-11142 | Bessel's Equation/x^2 y'' + x y' + (x^2 - (1 over 4)) y = 0 | The special case of Bessel's equation:
:$(1): \quad x^2 y' ' + x y' + \paren {x^2 - \dfrac 1 4} y = 0$
has the general solution:
:$y = C_1 \dfrac {\sin x} {\sqrt x} + C_2 \dfrac {\cos x} {\sqrt x}$ | === Particular Solution ===
{{:Bessel's Equation/x^2 y'' + x y' + (x^2 - (1 over 4)) y = 0/Particular Solution}}{{qed|lemma}}
$(1)$ can be expressed as:
:$(2): \quad y' ' + \dfrac 1 x y' + \paren {1 - \dfrac 1 {4 x^2} } y = 0$
which is in the form:
:$y' ' + \map P x y' + \map Q x y = 0$
where:
:$\map P x = \dfrac 1 x$
... | The special case of [[Definition:Bessel's Equation|Bessel's equation]]:
:$(1): \quad x^2 y' ' + x y' + \paren {x^2 - \dfrac 1 4} y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 \dfrac {\sin x} {\sqrt x} + C_2 \dfrac {\cos x} {\sqrt x}$ | === [[Bessel's Equation/x^2 y'' + x y' + (x^2 - (1 over 4)) y = 0/Particular Solution|Particular Solution]] ===
{{:Bessel's Equation/x^2 y'' + x y' + (x^2 - (1 over 4)) y = 0/Particular Solution}}{{qed|lemma}}
$(1)$ can be expressed as:
:$(2): \quad y' ' + \dfrac 1 x y' + \paren {1 - \dfrac 1 {4 x^2} } y = 0$
which i... | Bessel's Equation/x^2 y'' + x y' + (x^2 - (1 over 4)) y = 0 | https://proofwiki.org/wiki/Bessel's_Equation/x^2_y''_+_x_y'_+_(x^2_-_(1_over_4))_y_=_0 | https://proofwiki.org/wiki/Bessel's_Equation/x^2_y''_+_x_y'_+_(x^2_-_(1_over_4))_y_=_0 | [
"Examples of Bessel's Equation"
] | [
"Definition:Bessel's Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Bessel's Equation/x^2 y'' + x y' + (x^2 - (1 over 4)) y = 0/Particular Solution",
"Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another",
"Definition:Differential Equation/Solution/Particular Solution",
"Primitive of Reciprocal",
"Primitive of Square of Cosecant Function",
"Tw... |
proofwiki-11143 | Bessel's Equation/x^2 y'' + x y' + (x^2 - (1 over 4)) y = 0/Particular Solution | The special case of Bessel's equation:
:$(1): \quad x^2 y' ' + x y' + \paren {x^2 - \dfrac 1 4} y = 0$
has a particular solution:
:$y = \dfrac {\sin x} {\sqrt x}$ | Note that:
{{begin-eqn}}
{{eqn | l = y_1
| r = \frac {\sin x} {\sqrt x}
| c =
}}
{{eqn | r = x^{-1/2} \sin x
| c =
}}
{{eqn | ll= \leadsto
| l = {y_1}'
| r = x^{-1/2} \cos x - \frac 1 2 x^{-3/2} \sin x
| c = Product Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = {y_1}' '... | The special case of [[Definition:Bessel's Equation|Bessel's equation]]:
:$(1): \quad x^2 y' ' + x y' + \paren {x^2 - \dfrac 1 4} y = 0$
has a [[Definition:Particular Solution|particular solution]]:
:$y = \dfrac {\sin x} {\sqrt x}$ | Note that:
{{begin-eqn}}
{{eqn | l = y_1
| r = \frac {\sin x} {\sqrt x}
| c =
}}
{{eqn | r = x^{-1/2} \sin x
| c =
}}
{{eqn | ll= \leadsto
| l = {y_1}'
| r = x^{-1/2} \cos x - \frac 1 2 x^{-3/2} \sin x
| c = [[Product Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = {y_... | Bessel's Equation/x^2 y'' + x y' + (x^2 - (1 over 4)) y = 0/Particular Solution | https://proofwiki.org/wiki/Bessel's_Equation/x^2_y''_+_x_y'_+_(x^2_-_(1_over_4))_y_=_0/Particular_Solution | https://proofwiki.org/wiki/Bessel's_Equation/x^2_y''_+_x_y'_+_(x^2_-_(1_over_4))_y_=_0/Particular_Solution | [
"Examples of Bessel's Equation"
] | [
"Definition:Bessel's Equation",
"Definition:Differential Equation/Solution/Particular Solution"
] | [
"Product Rule for Derivatives",
"Product Rule for Derivatives",
"Definition:Differential Equation/Solution/Particular Solution"
] |
proofwiki-11144 | Linear Second Order ODE/(x - 1) y'' - y' + y = 0 | The second order ODE:
:$(1): \quad \paren {x - 1} y' ' - x y' + y = 0$
has the general solution:
:$y = C_1 x + C_2 e^x$ | Note that:
{{begin-eqn}}
{{eqn | l = y_1
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = {y_1}'
| r = 1
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = {y_1}' '
| r = 0
| c = Derivative of Constant
}}
{{end-eqn}}
and so by inspection:
:$y_1 = x$
is a particula... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad \paren {x - 1} y' ' - x y' + y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 x + C_2 e^x$ | Note that:
{{begin-eqn}}
{{eqn | l = y_1
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = {y_1}'
| r = 1
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = {y_1}' '
| r = 0
| c = [[Derivative of Constant]]
}}
{{end-eqn}}
and so by inspection:
:$y_1 = x$
is a... | Linear Second Order ODE/(x - 1) y'' - y' + y = 0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/(x_-_1)_y''_-_y'_+_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/(x_-_1)_y''_-_y'_+_y_=_0 | [
"Examples of Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Power Rule for Derivatives",
"Derivative of Constant",
"Definition:Differential Equation/Solution/Particular Solution",
"Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another",
"Definition:Differential Equation/Solution/Particular Solution",
"Primitive of x over a x + b",
"Pr... |
proofwiki-11145 | Linear Second Order ODE/x^2 y'' + 2 x y' - 2 y = 0 | The second order ODE:
:$(1): \quad x^2 y'' + 2 x y' - 2 y = 0$
has the general solution:
:$y = C_1 x + \dfrac {C_2} {x^2}$ | Note that:
{{begin-eqn}}
{{eqn | l = y_1
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = {y_1}'
| r = 1
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = {y_1}''
| r = 0
| c = Derivative of Constant
}}
{{end-eqn}}
and so by inspection:
:$y_1 = x$
is a particular... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad x^2 y'' + 2 x y' - 2 y = 0$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$y = C_1 x + \dfrac {C_2} {x^2}$ | Note that:
{{begin-eqn}}
{{eqn | l = y_1
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = {y_1}'
| r = 1
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = {y_1}''
| r = 0
| c = [[Derivative of Constant]]
}}
{{end-eqn}}
and so by inspection:
:$y_1 = x$
is a ... | Linear Second Order ODE/x^2 y'' + 2 x y' - 2 y = 0/Proof 1 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/x^2_y''_+_2_x_y'_-_2_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/x^2_y''_+_2_x_y'_-_2_y_=_0/Proof_1 | [
"Examples of Cauchy-Euler Equation",
"Linear Second Order ODE/x^2 y'' + 2 x y' - 2 y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Power Rule for Derivatives",
"Derivative of Constant",
"Definition:Differential Equation/Solution/Particular Solution",
"Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another",
"Definition:Differential Equation/Solution/Particular Solution",
"Primitive of Reciprocal",
"Logari... |
proofwiki-11146 | Linear Second Order ODE/x^2 y'' + 2 x y' - 2 y = 0 | The second order ODE:
:$(1): \quad x^2 y'' + 2 x y' - 2 y = 0$
has the general solution:
:$y = C_1 x + \dfrac {C_2} {x^2}$ | It can be seen that $(1)$ is an instance of the Cauchy-Euler Equation:
:$x^2 y'' + p x y' + q y = 0$
where:
:$p = 2$
:$q = -2$
By Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE, this can be expressed as:
:$\dfrac {\d^2 y} {\d t^2} + \paren {p - 1} \dfrac {\d y} {\d t} + q y = 0$
by making the su... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad x^2 y'' + 2 x y' - 2 y = 0$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$y = C_1 x + \dfrac {C_2} {x^2}$ | It can be seen that $(1)$ is an instance of the [[Definition:Cauchy-Euler Equation|Cauchy-Euler Equation]]:
:$x^2 y'' + p x y' + q y = 0$
where:
:$p = 2$
:$q = -2$
By [[Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE]], this can be expressed as:
:$\dfrac {\d^2 y} {\d t^2} + \paren {p - 1} \dfra... | Linear Second Order ODE/x^2 y'' + 2 x y' - 2 y = 0/Proof 2 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/x^2_y''_+_2_x_y'_-_2_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/x^2_y''_+_2_x_y'_-_2_y_=_0/Proof_2 | [
"Examples of Cauchy-Euler Equation",
"Linear Second Order ODE/x^2 y'' + 2 x y' - 2 y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Cauchy-Euler Equation",
"Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE",
"Linear Second Order ODE/y'' + y' - 2 y = 0",
"Definition:Differential Equation/Solution/General Solution"
] |
proofwiki-11147 | Second Order ODE/y'' - x f(x) y' + f(x) y = 0 | The second order ODE:
:$(1): \quad y' ' - x \, \map f x y' + \map f x y = 0$
has the general solution:
:$\ds y = C_1 x + C_2 x \int x^{-2} e^{\int x \, \map f x \rd x} \rd x$ | Note that:
{{begin-eqn}}
{{eqn | l = y_1
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = {y_1}'
| r = 1
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = {y_1}' '
| r = 0
| c = Derivative of Constant
}}
{{end-eqn}}
Substituting into $(1)$:
{{begin-eqn}}
{{eqn | ... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' - x \, \map f x y' + \map f x y = 0$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$\ds y = C_1 x + C_2 x \int x^{-2} e^{\int x \, \map f x \rd x} \rd x$ | Note that:
{{begin-eqn}}
{{eqn | l = y_1
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = {y_1}'
| r = 1
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = {y_1}' '
| r = 0
| c = [[Derivative of Constant]]
}}
{{end-eqn}}
Substituting into $(1)$:
{{begin-eq... | Second Order ODE/y'' - x f(x) y' + f(x) y = 0 | https://proofwiki.org/wiki/Second_Order_ODE/y''_-_x_f(x)_y'_+_f(x)_y_=_0 | https://proofwiki.org/wiki/Second_Order_ODE/y''_-_x_f(x)_y'_+_f(x)_y_=_0 | [
"Examples of Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Power Rule for Derivatives",
"Derivative of Constant",
"Definition:Differential Equation/Solution/Particular Solution",
"Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another",
"Definition:Differential Equation/Solution/Particular Solution",
"Two Linearly Independent Solutions ... |
proofwiki-11148 | Second Order ODE/x y'' - (2 x + 1) y' + (x + 1) y = 0 | The second order ODE:
:$(1): \quad x y' ' - \paren {2 x + 1} y' + \paren {x + 1} y = 0$
has the general solution:
:$y = C_1 e^x + C_2 x^2 e^x$ | Note that:
:$x - \paren {2 x + 1} + \paren {x + 1} = 0$
so if $y' ' = y' = y$ we find that $(1)$ is satisfied.
So:
{{begin-eqn}}
{{eqn | l = y_1
| r = e^x
| c =
}}
{{eqn | ll= \leadsto
| l = {y_1}'
| r = e^x
| c = Derivative of Exponential Function
}}
{{eqn | ll= \leadsto
| l = {y_1... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad x y' ' - \paren {2 x + 1} y' + \paren {x + 1} y = 0$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$y = C_1 e^x + C_2 x^2 e^x$ | Note that:
:$x - \paren {2 x + 1} + \paren {x + 1} = 0$
so if $y' ' = y' = y$ we find that $(1)$ is satisfied.
So:
{{begin-eqn}}
{{eqn | l = y_1
| r = e^x
| c =
}}
{{eqn | ll= \leadsto
| l = {y_1}'
| r = e^x
| c = [[Derivative of Exponential Function]]
}}
{{eqn | ll= \leadsto
| l ... | Second Order ODE/x y'' - (2 x + 1) y' + (x + 1) y = 0 | https://proofwiki.org/wiki/Second_Order_ODE/x_y''_-_(2_x_+_1)_y'_+_(x_+_1)_y_=_0 | https://proofwiki.org/wiki/Second_Order_ODE/x_y''_-_(2_x_+_1)_y'_+_(x_+_1)_y_=_0 | [
"Examples of Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Derivative of Exponential Function",
"Derivative of Exponential Function",
"Definition:Differential Equation/Solution/Particular Solution",
"Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another",
"Definition:Differential Equation/Solution/Particular Solution",
"Two Linearly In... |
proofwiki-11149 | Second Order ODE/y'' - f(x) y' + (f(x) - 1) y = 0 | The second order ODE:
:$(1): \quad y' ' - \map f x y' + \paren {\map f x - 1} y = 0$
has the general solution:
:$\ds y = C_1 e^x + C_2 e^x \int e^{-2 x + \int \map f x \rd x} \rd x$ | Note that:
:$1 - \map f x + \paren {\map f x - 1} = 0$
so if $y' ' = y' = y$ we find that $(1)$ is satisfied.
So:
{{begin-eqn}}
{{eqn | l = y_1
| r = e^x
| c =
}}
{{eqn | ll= \leadsto
| l = {y_1}'
| r = e^x
| c = Derivative of Exponential Function
}}
{{eqn | ll= \leadsto
| l = {y_1}... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' - \map f x y' + \paren {\map f x - 1} y = 0$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$\ds y = C_1 e^x + C_2 e^x \int e^{-2 x + \int \map f x \rd x} \rd x$ | Note that:
:$1 - \map f x + \paren {\map f x - 1} = 0$
so if $y' ' = y' = y$ we find that $(1)$ is satisfied.
So:
{{begin-eqn}}
{{eqn | l = y_1
| r = e^x
| c =
}}
{{eqn | ll= \leadsto
| l = {y_1}'
| r = e^x
| c = [[Derivative of Exponential Function]]
}}
{{eqn | ll= \leadsto
| l =... | Second Order ODE/y'' - f(x) y' + (f(x) - 1) y = 0 | https://proofwiki.org/wiki/Second_Order_ODE/y''_-_f(x)_y'_+_(f(x)_-_1)_y_=_0 | https://proofwiki.org/wiki/Second_Order_ODE/y''_-_f(x)_y'_+_(f(x)_-_1)_y_=_0 | [
"Examples of Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Derivative of Exponential Function",
"Derivative of Exponential Function",
"Definition:Differential Equation/Solution/Particular Solution",
"Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another",
"Definition:Differential Equation/Solution/Particular Solution",
"Two Linearly In... |
proofwiki-11150 | Exponential Function is Solution of Constant Coefficient Homogeneous LSOODE iff Index is Root of Auxiliary Equation | Let:
:$(1): \quad y' ' + p y' + q y = 0$
be a constant coefficient homogeneous linear second order ODE.
Then:
:$y = e^{m_1 x}$ is a solution to $(1)$
{{iff}}
:$m_1$ is a root of the auxiliary equation $m^2 + p m + q = 0$ | Consider the equation:
:$y = e^{m_1 x}$
Differentiating {{WRT|Differentiation}} $x$:
{{begin-eqn}}
{{eqn | l = y
| r = e^{m_1 x}
| c =
}}
{{eqn | l = y'
| r = m_1 e^{m_1 x}
| c =
}}
{{eqn | l = y' '
| r = {m_1}^2 e^{m_1 x}
| c =
}}
{{end-eqn}} | Let:
:$(1): \quad y' ' + p y' + q y = 0$
be a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Then:
:$y = e^{m_1 x}$ is a [[Definition:Solution to Differential Equation|solution]] to $(1)$
{{iff}}
:$m_1$ is a [[Definition:Root of Polyno... | Consider the equation:
:$y = e^{m_1 x}$
[[Definition:Differentiation|Differentiating]] {{WRT|Differentiation}} $x$:
{{begin-eqn}}
{{eqn | l = y
| r = e^{m_1 x}
| c =
}}
{{eqn | l = y'
| r = m_1 e^{m_1 x}
| c =
}}
{{eqn | l = y' '
| r = {m_1}^2 e^{m_1 x}
| c =
}}
{{end-eqn}} | Exponential Function is Solution of Constant Coefficient Homogeneous LSOODE iff Index is Root of Auxiliary Equation | https://proofwiki.org/wiki/Exponential_Function_is_Solution_of_Constant_Coefficient_Homogeneous_LSOODE_iff_Index_is_Root_of_Auxiliary_Equation | https://proofwiki.org/wiki/Exponential_Function_is_Solution_of_Constant_Coefficient_Homogeneous_LSOODE_iff_Index_is_Root_of_Auxiliary_Equation | [
"Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Differential Equation/Solution",
"Definition:Root of Polynomial",
"Definition:Auxiliary Equation"
] | [
"Definition:Differentiation"
] |
proofwiki-11151 | Solution of Constant Coefficient Homogeneous LSOODE | Let:
:$(1): \quad y' ' + p y' + q y = 0$
be a constant coefficient homogeneous linear second order ODE.
Let $m_1$ and $m_2$ be the roots of the auxiliary equation $m^2 + p m + q = 0$. | === Real Roots of Auxiliary Equation ===
{{:Solution of Constant Coefficient Homogeneous LSOODE/Real Roots of Auxiliary Equation}}{{qed|lemma}} | Let:
:$(1): \quad y' ' + p y' + q y = 0$
be a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Let $m_1$ and $m_2$ be the [[Definition:Root of Polynomial|roots]] of the [[Definition:Auxiliary Equation|auxiliary equation]] $m^2 + p m + q =... | === [[Solution of Constant Coefficient Homogeneous LSOODE/Real Roots of Auxiliary Equation|Real Roots of Auxiliary Equation]] ===
{{:Solution of Constant Coefficient Homogeneous LSOODE/Real Roots of Auxiliary Equation}}{{qed|lemma}} | Solution of Constant Coefficient Homogeneous LSOODE | https://proofwiki.org/wiki/Solution_of_Constant_Coefficient_Homogeneous_LSOODE | https://proofwiki.org/wiki/Solution_of_Constant_Coefficient_Homogeneous_LSOODE | [
"Solution of Constant Coefficient Homogeneous LSOODE",
"Constant Coefficient Homogeneous LSOODEs",
"Ordinary Differential Equations",
"Differential Equations"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Root of Polynomial",
"Definition:Auxiliary Equation"
] | [
"Solution of Constant Coefficient Homogeneous LSOODE/Real Roots of Auxiliary Equation"
] |
proofwiki-11152 | Solution of Constant Coefficient Homogeneous LSOODE/Real Roots of Auxiliary Equation | {{:Solution of Constant Coefficient Homogeneous LSOODE}}
Let $p^2 > 4 q$.
Then $(1)$ has the general solution:
:$y = C_1 e^{m_1 x} + C_2 e^{m_2 x}$ | Consider the auxiliary equation of $(1)$:
:$(2): \quad m^2 + p m + q$
Let $p^2 > 4 q$.
From Solution to Quadratic Equation with Real Coefficients, $(2)$ has two real roots:
{{begin-eqn}}
{{eqn | l = m_1
| r = -\frac p 2 + \sqrt {\frac {p^2} 4 - q}
}}
{{eqn | l = m_2
| r = -\frac p 2 - \sqrt {\frac {p^2} 4 -... | {{:Solution of Constant Coefficient Homogeneous LSOODE}}
Let $p^2 > 4 q$.
Then $(1)$ has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^{m_1 x} + C_2 e^{m_2 x}$ | Consider the [[Definition:Auxiliary Equation|auxiliary equation]] of $(1)$:
:$(2): \quad m^2 + p m + q$
Let $p^2 > 4 q$.
From [[Solution to Quadratic Equation with Real Coefficients]], $(2)$ has two [[Definition:Real Number|real]] [[Definition:Root of Polynomial|roots]]:
{{begin-eqn}}
{{eqn | l = m_1
| r = -\... | Solution of Constant Coefficient Homogeneous LSOODE/Real Roots of Auxiliary Equation | https://proofwiki.org/wiki/Solution_of_Constant_Coefficient_Homogeneous_LSOODE/Real_Roots_of_Auxiliary_Equation | https://proofwiki.org/wiki/Solution_of_Constant_Coefficient_Homogeneous_LSOODE/Real_Roots_of_Auxiliary_Equation | [
"Solution of Constant Coefficient Homogeneous LSOODE"
] | [
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Real Number",
"Definition:Root of Polynomial",
"Exponential Function is Solution of Constant Coefficient Homogeneous LSOODE iff Index is Root of Auxiliary Equation",
"Definition:Differential Equation/Solution/... |
proofwiki-11153 | Solution of Constant Coefficient Homogeneous LSOODE/Complex Roots of Auxiliary Equation | {{:Solution of Constant Coefficient Homogeneous LSOODE}}
Let $p^2 < 4 q$.
Then $(1)$ has the general solution:
:$y = e^{a x} \paren {C_1 \cos b x + C_2 \sin b x}$
where:
:$m_1 = a + i b$
:$m_2 = a - i b$ | Consider the auxiliary equation of $(1)$:
:$(2): \quad m^2 + p m + q$
Let $p^2 < 4 q$.
From Solution to Quadratic Equation with Real Coefficients, $(2)$ has two complex roots:
{{begin-eqn}}
{{eqn | l = m_1
| r = -\frac p 2 + i \sqrt {q - \frac {p^2} 4}
}}
{{eqn | l = m_2
| r = -\frac p 2 - i \sqrt {q - \fra... | {{:Solution of Constant Coefficient Homogeneous LSOODE}}
Let $p^2 < 4 q$.
Then $(1)$ has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = e^{a x} \paren {C_1 \cos b x + C_2 \sin b x}$
where:
:$m_1 = a + i b$
:$m_2 = a - i b$ | Consider the [[Definition:Auxiliary Equation|auxiliary equation]] of $(1)$:
:$(2): \quad m^2 + p m + q$
Let $p^2 < 4 q$.
From [[Solution to Quadratic Equation with Real Coefficients]], $(2)$ has two [[Definition:Complex Number|complex]] [[Definition:Root of Polynomial|roots]]:
{{begin-eqn}}
{{eqn | l = m_1
| ... | Solution of Constant Coefficient Homogeneous LSOODE/Complex Roots of Auxiliary Equation | https://proofwiki.org/wiki/Solution_of_Constant_Coefficient_Homogeneous_LSOODE/Complex_Roots_of_Auxiliary_Equation | https://proofwiki.org/wiki/Solution_of_Constant_Coefficient_Homogeneous_LSOODE/Complex_Roots_of_Auxiliary_Equation | [
"Solution of Constant Coefficient Homogeneous LSOODE"
] | [
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Complex Number",
"Definition:Root of Polynomial",
"Exponential Function is Solution of Constant Coefficient Homogeneous LSOODE iff Index is Root of Auxiliary Equation",
"Definition:Differential Equation/Soluti... |
proofwiki-11154 | Solution of Constant Coefficient Homogeneous LSOODE/Equal Real Roots of Auxiliary Equation | {{:Solution of Constant Coefficient Homogeneous LSOODE}}
Let $p^2 = 4 q$.
Then $(1)$ has the general solution:
:$y = C_1 e^{m_1 x} + C_2 x e^{m_1 x}$ | Consider the auxiliary equation of $(1)$:
:$(2): \quad m^2 + p m + q$
Let $p^2 = 4 q$.
From Solution to Quadratic Equation with Real Coefficients, $(2)$ has one (repeated) root, that is:
:$m_1 = m_2 = -\dfrac p 2$
From Exponential Function is Solution of Constant Coefficient Homogeneous LSOODE iff Index is Root of Auxi... | {{:Solution of Constant Coefficient Homogeneous LSOODE}}
Let $p^2 = 4 q$.
Then $(1)$ has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^{m_1 x} + C_2 x e^{m_1 x}$ | Consider the [[Definition:Auxiliary Equation|auxiliary equation]] of $(1)$:
:$(2): \quad m^2 + p m + q$
Let $p^2 = 4 q$.
From [[Solution to Quadratic Equation with Real Coefficients]], $(2)$ has one (repeated) [[Definition:Root of Polynomial|root]], that is:
:$m_1 = m_2 = -\dfrac p 2$
From [[Exponential Function ... | Solution of Constant Coefficient Homogeneous LSOODE/Equal Real Roots of Auxiliary Equation | https://proofwiki.org/wiki/Solution_of_Constant_Coefficient_Homogeneous_LSOODE/Equal_Real_Roots_of_Auxiliary_Equation | https://proofwiki.org/wiki/Solution_of_Constant_Coefficient_Homogeneous_LSOODE/Equal_Real_Roots_of_Auxiliary_Equation | [
"Solution of Constant Coefficient Homogeneous LSOODE"
] | [
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Exponential Function is Solution of Constant Coefficient Homogeneous LSOODE iff Index is Root of Auxiliary Equation",
"Definition:Differential Equation/Solution/Particular Solution",
"Par... |
proofwiki-11155 | Linear Second Order ODE/y'' + y' - 6 y = 0 | The second order ODE:
:$(1): \quad y' ' + y' - 6 y = 0$
has the general solution:
:$y = C_1 e^{2 x} + C_2 e^{-3 x}$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad: m^2 + m - 6 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = -3$
:$m_2 = 2$
These are real and unequal.
So from Solution of Constant Coefficie... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' + y' - 6 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^{2 x} + C_2 e^{-3 x}$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad: m^2 + m - 6 = 0$
From [[Solution to Quadratic Equation with Real Coefficients]], t... | Linear Second Order ODE/y'' + y' - 6 y = 0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_y'_-_6_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_y'_-_6_y_=_0 | [
"Examples of Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Definition:Real Number",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential Equ... |
proofwiki-11156 | Linear Second Order ODE/y'' + 2 y' + y = 0 | The second order ODE:
:$(1): \quad y' ' + 2 y' + y = 0$
has the general solution:
:$y = C_1 e^{-x} + C_2 x e^{-x}$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad m^2 + 2 m + 1 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = m_2 = -1$
These are real and equal.
So from Solution of Constant Coefficient Hom... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' + 2 y' + y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^{-x} + C_2 x e^{-x}$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad m^2 + 2 m + 1 = 0$
From [[Solution to Quadratic Equation with Real Coefficients]], ... | Linear Second Order ODE/y'' + 2 y' + y = 0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_2_y'_+_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_2_y'_+_y_=_0 | [
"Examples of Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Definition:Real Number",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential Equ... |
proofwiki-11157 | Linear Second Order ODE/y'' + 8 y = 0 | The second order ODE:
:$(1): \quad y' ' + 8 y = 0$
has the general solution:
:$y = C_1 \cos 2 \sqrt 2 x + C_2 \sin 2 \sqrt 2 x$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad: m^2 + 8 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = 2 \sqrt 2 i$
:$m_2 = -2 \sqrt 2 i$
These are complex and unequal.
So from Solution of... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' + 8 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 \cos 2 \sqrt 2 x + C_2 \sin 2 \sqrt 2 x$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad: m^2 + 8 = 0$
From [[Solution to Quadratic Equation with Real Coefficients]], the [... | Linear Second Order ODE/y'' + 8 y = 0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_8_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_8_y_=_0 | [
"Examples of Linear Second Order ODE/y'' + k^2 y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Definition:Complex Number",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential ... |
proofwiki-11158 | Linear Second Order ODE/2 y'' - 4 y' + 8 y = 0 | The second order ODE:
:$(1): \quad 2 y' ' - 4 y + 8 y = 0$
has the general solution:
:$y = e^x \paren {C_1 \cos \sqrt 3 x + C_2 \sin \sqrt 3 x}$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Let $(1)$ be written in the form:
:$y' ' - 2 y + 4 y = 0$
Its auxiliary equation is:
:$(2): \quad: m^2 - 2 m + 4 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = 1 + \sqrt 3 i$
:$m_2 ... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad 2 y' ' - 4 y + 8 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = e^x \paren {C_1 \cos \sqrt 3 x + C_2 \sin \sqrt 3 x}$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Let $(1)$ be written in the form:
:$y' ' - 2 y + 4 y = 0$
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad: m^2 - 2 m + 4 = 0$
From... | Linear Second Order ODE/2 y'' - 4 y' + 8 y = 0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/2_y''_-_4_y'_+_8_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/2_y''_-_4_y'_+_8_y_=_0 | [
"Examples of Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Definition:Complex Number",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential ... |
proofwiki-11159 | Linear Second Order ODE/y'' - 9 y' + 20 y = 0 | The second order ODE:
:$(1): \quad y' ' - 9 y' + 20 y = 0$
has the general solution:
:$y = C_1 e^{4 x} + C_2 e^{5 x}$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad: m^2 - 9 m + 20 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = 4$
:$m_2 = 5$
These are real and unequal.
So from Solution of Constant Coeffic... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' - 9 y' + 20 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^{4 x} + C_2 e^{5 x}$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad: m^2 - 9 m + 20 = 0$
From [[Solution to Quadratic Equation with Real Coefficients]]... | Linear Second Order ODE/y'' - 9 y' + 20 y = 0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_9_y'_+_20_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_9_y'_+_20_y_=_0 | [
"Examples of Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Definition:Real Number",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential Equ... |
proofwiki-11160 | Linear Second Order ODE/2 y'' + 2 y' + 3 y = 0 | The second order ODE:
:$(1): \quad 2 y' ' + 2 y' + 3 y = 0$
has the general solution:
:$y = e^{-x/2} \paren {C_1 \cos \dfrac {\sqrt 5} 2 x + C_2 \sin \dfrac {\sqrt 5} 2 x}$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad: 2 m^2 + 2 m + 3 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = -\dfrac 1 2 + \dfrac {\sqrt 5} 2 i$
:$m_2 = -\dfrac 1 2 - \dfrac {\sqrt 5} 2 ... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad 2 y' ' + 2 y' + 3 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = e^{-x/2} \paren {C_1 \cos \dfrac {\sqrt 5} 2 x + C_2 \sin \dfrac {\sqrt 5} 2 x}$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad: 2 m^2 + 2 m + 3 = 0$
From [[Solution to Quadratic Equation with Real Coefficients]... | Linear Second Order ODE/2 y'' + 2 y' + 3 y = 0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/2_y''_+_2_y'_+_3_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/2_y''_+_2_y'_+_3_y_=_0 | [
"Examples of Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Definition:Complex Number",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential ... |
proofwiki-11161 | Condition for Solutions to Constant Coefficient Homogeneous LSOODE to tend to Zero | Let:
:$(1): \quad y' ' + p y' + q y = 0$
be a constant coefficient homogeneous linear second order ODE.
Let the general solution to $(1)$ be $\map y {x, C_1, C_2}$.
Then:
:$\ds \lim_{x \mathop \to \infty} \map y {x, C_1, C_2} = 0$
{{iff}}
:$p$ and $q$ are both strictly positive. | By Solution of Constant Coefficient Homogeneous LSOODE, $y$ is in one of the following forms:
:<nowiki>$y = \begin{cases}
C_1 e^{m_1 x} + C_2 e^{m_2 x} & : p^2 > 4 q \\
& \\
C_1 e^{m_1 x} + C_2 x e^{m_2 x} & : p^2 = 4 q \\
& \\
e^{a x} \paren {C_1 \sin b x + C_2 \cos b x} & : p^2 < 4 q
\end{cases}$</nowiki>
where:
:$... | Let:
:$(1): \quad y' ' + p y' + q y = 0$
be a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Let the [[Definition:General Solution to Differential Equation|general solution]] to $(1)$ be $\map y {x, C_1, C_2}$.
Then:
:$\ds \lim_{x \mat... | By [[Solution of Constant Coefficient Homogeneous LSOODE]], $y$ is in one of the following forms:
:<nowiki>$y = \begin{cases}
C_1 e^{m_1 x} + C_2 e^{m_2 x} & : p^2 > 4 q \\
& \\
C_1 e^{m_1 x} + C_2 x e^{m_2 x} & : p^2 = 4 q \\
& \\
e^{a x} \paren {C_1 \sin b x + C_2 \cos b x} & : p^2 < 4 q
\end{cases}$</nowiki>
where... | Condition for Solutions to Constant Coefficient Homogeneous LSOODE to tend to Zero | https://proofwiki.org/wiki/Condition_for_Solutions_to_Constant_Coefficient_Homogeneous_LSOODE_to_tend_to_Zero | https://proofwiki.org/wiki/Condition_for_Solutions_to_Constant_Coefficient_Homogeneous_LSOODE_to_tend_to_Zero | [
"Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Strictly Positive/Real Number"
] | [
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Root of Polynomial",
"Definition:Auxiliary Equation",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Solution of Constant Coefficient Homogeneous LSOODE"
] |
proofwiki-11162 | Derivative of Solution to Constant Coefficient Homogeneous LSOODE is also Solution | Let:
:$(1): \quad y' ' + p y' + q y = 0$
be a constant coefficient homogeneous linear second order ODE.
Let $\map y x$ be a particular solution of $(1)$.
Then its derivative $\map {y'} x$ is also a particular solution of $(1)$. | By Solution of Constant Coefficient Homogeneous LSOODE, $y$ is in one of the following forms:
:<nowiki>$y = \begin{cases}
C_1 e^{m_1 x} + C_2 e^{m_2 x} & : p^2 > 4 q \\
& \\
C_1 e^{m_1 x} + C_2 x e^{m_2 x} & : p^2 = 4 q \\
& \\
e^{a x} \paren {C_1 \sin b x + C_2 \cos b x} & : p^2 < 4 q
\end{cases}$</nowiki>
Let:
:$y ... | Let:
:$(1): \quad y' ' + p y' + q y = 0$
be a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Let $\map y x$ be a [[Definition:Particular Solution|particular solution]] of $(1)$.
Then its [[Definition:Derivative|derivative]] $\map {y'}... | By [[Solution of Constant Coefficient Homogeneous LSOODE]], $y$ is in one of the following forms:
:<nowiki>$y = \begin{cases}
C_1 e^{m_1 x} + C_2 e^{m_2 x} & : p^2 > 4 q \\
& \\
C_1 e^{m_1 x} + C_2 x e^{m_2 x} & : p^2 = 4 q \\
& \\
e^{a x} \paren {C_1 \sin b x + C_2 \cos b x} & : p^2 < 4 q
\end{cases}$</nowiki>
Let... | Derivative of Solution to Constant Coefficient Homogeneous LSOODE is also Solution | https://proofwiki.org/wiki/Derivative_of_Solution_to_Constant_Coefficient_Homogeneous_LSOODE_is_also_Solution | https://proofwiki.org/wiki/Derivative_of_Solution_to_Constant_Coefficient_Homogeneous_LSOODE_is_also_Solution | [
"Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Derivative",
"Definition:Differential Equation/Solution/Particular Solution"
] | [
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Differential Equation/Solution/Particular Solution"
] |
proofwiki-11163 | Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE | Consider the Cauchy-Euler equation:
:$(1): \quad x^2 \dfrac {\d^2 y} {\d x^2} + p x \dfrac {\d y} {\d x} + q y = 0$
By making the substitution:
:$x = e^t$
it is possible to convert $(1)$ into a constant coefficient homogeneous linear second order ODE:
:$\dfrac {\d^2 y} {\d t^2} + \paren {p - 1} \dfrac {\d y} {\d t} + q... | We have:
{{begin-eqn}}
{{eqn | l = x
| r = e^t
| c = Derivative of Exponential Function
}}
{{eqn | ll=\leadsto
| l = \frac {\d x} {\d t}
| r = e^t
| c =
}}
{{eqn | n = 1
| r = x
| c =
}}
{{eqn | ll=\leadsto
| l = \frac {\d t} {\d x}
| r = \frac 1 x
| c = Der... | Consider the [[Definition:Cauchy-Euler Equation|Cauchy-Euler equation]]:
:$(1): \quad x^2 \dfrac {\d^2 y} {\d x^2} + p x \dfrac {\d y} {\d x} + q y = 0$
By making the substitution:
:$x = e^t$
it is possible to convert $(1)$ into a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coeffici... | We have:
{{begin-eqn}}
{{eqn | l = x
| r = e^t
| c = [[Derivative of Exponential Function]]
}}
{{eqn | ll=\leadsto
| l = \frac {\d x} {\d t}
| r = e^t
| c =
}}
{{eqn | n = 1
| r = x
| c =
}}
{{eqn | ll=\leadsto
| l = \frac {\d t} {\d x}
| r = \frac 1 x
| c ... | Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE | https://proofwiki.org/wiki/Conversion_of_Cauchy-Euler_Equation_to_Constant_Coefficient_Linear_ODE | https://proofwiki.org/wiki/Conversion_of_Cauchy-Euler_Equation_to_Constant_Coefficient_Linear_ODE | [
"Cauchy-Euler Equation"
] | [
"Definition:Cauchy-Euler Equation",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients"
] | [
"Derivative of Exponential Function",
"Derivative of Inverse Function",
"Derivative of Composite Function",
"Derivative of Composite Function",
"Product Rule for Derivatives"
] |
proofwiki-11164 | Linear Second Order ODE/x^2 y'' + 3 x y' + 10 y = 0 | The second order ODE:
:$(1): \quad x^2 y'' + 3 x y' + 10 y = 0$
has the general solution:
:$y = \dfrac 1 x \paren {C_1 \map \cos {\ln x^3} + C_2 \map \sin {\ln x^3} }$ | $(1)$ is an instance of the Cauchy-Euler Equation:
:$x^2 y'' + p x y' + q y = 0$
where:
:$p = 3$
:$q = 10$
By Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE, this can be expressed as:
:$\dfrac {\d^2 y} {\d t^2} + \paren {p - 1} \dfrac {\d y} {\d t^2} + q y = 0$
by making the substitution:
:$x = ... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad x^2 y'' + 3 x y' + 10 y = 0$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$y = \dfrac 1 x \paren {C_1 \map \cos {\ln x^3} + C_2 \map \sin {\ln x^3} }$ | $(1)$ is an instance of the [[Definition:Cauchy-Euler Equation|Cauchy-Euler Equation]]:
:$x^2 y'' + p x y' + q y = 0$
where:
:$p = 3$
:$q = 10$
By [[Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE]], this can be expressed as:
:$\dfrac {\d^2 y} {\d t^2} + \paren {p - 1} \dfrac {\d y} {\d t^2} + ... | Linear Second Order ODE/x^2 y'' + 3 x y' + 10 y = 0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/x^2_y''_+_3_x_y'_+_10_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/x^2_y''_+_3_x_y'_+_10_y_=_0 | [
"Examples of Cauchy-Euler Equation"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Cauchy-Euler Equation",
"Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' + 2 y' + 10 y = 0",
"Definition:Differential Equation/Solution/General Solution"
] |
proofwiki-11165 | Linear Second Order ODE/2 x^2 y'' + 10 x y' + 8 y = 0 | The second order ODE:
:$(1): \quad 2 x^2 y' ' + 10 x y' + 8 y = 0$
has the general solution:
:$y = C_1 x^{-2} + C_2 x^{-2} \ln x$ | Let $(1)$ be rewritten as:
:$x^2 y' ' + 5 x y' + 4 y = 0$
It can be seen to be an instance of the Cauchy-Euler Equation:
:$x^2 y' ' + p x y' + q y = 0$
where:
:$p = 5$
:$q = 4$
By Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE, this can be expressed as:
:$\dfrac {\d^2 y} {\d t^2} + \paren {p - 1... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad 2 x^2 y' ' + 10 x y' + 8 y = 0$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$y = C_1 x^{-2} + C_2 x^{-2} \ln x$ | Let $(1)$ be rewritten as:
:$x^2 y' ' + 5 x y' + 4 y = 0$
It can be seen to be an instance of the [[Definition:Cauchy-Euler Equation|Cauchy-Euler Equation]]:
:$x^2 y' ' + p x y' + q y = 0$
where:
:$p = 5$
:$q = 4$
By [[Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE]], this can be expressed as... | Linear Second Order ODE/2 x^2 y'' + 10 x y' + 8 y = 0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/2_x^2_y''_+_10_x_y'_+_8_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/2_x^2_y''_+_10_x_y'_+_8_y_=_0 | [
"Examples of Cauchy-Euler Equation"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Cauchy-Euler Equation",
"Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE",
"Linear Second Order ODE/y'' + 4 y' + 4 y = 0",
"Definition:Differential Equation/Solution/General Solution"
] |
proofwiki-11166 | Linear Second Order ODE/x^2 y'' + 2 x y' - 12 y = 0 | The second order ODE:
:$(1): \quad x^2 y'' + 2 x y' - 12 y = 0$
has the general solution:
:$y = C_1 x^3 + C_2 x^{-4}$ | It can be seen that $(1)$ is an instance of the Cauchy-Euler Equation:
:$x^2 y'' + p x y' + q y = 0$
where:
:$p = 2$
:$q = -12$
By Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE, this can be expressed as:
:$\dfrac {\d^2 y} {\d t^2} + \paren {p - 1} \dfrac {\d y} {\d t^2} + q y = 0$
by making the... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad x^2 y'' + 2 x y' - 12 y = 0$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$y = C_1 x^3 + C_2 x^{-4}$ | It can be seen that $(1)$ is an instance of the [[Definition:Cauchy-Euler Equation|Cauchy-Euler Equation]]:
:$x^2 y'' + p x y' + q y = 0$
where:
:$p = 2$
:$q = -12$
By [[Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE]], this can be expressed as:
:$\dfrac {\d^2 y} {\d t^2} + \paren {p - 1} \dfr... | Linear Second Order ODE/x^2 y'' + 2 x y' - 12 y = 0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/x^2_y''_+_2_x_y'_-_12_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/x^2_y''_+_2_x_y'_-_12_y_=_0 | [
"Examples of Cauchy-Euler Equation"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Cauchy-Euler Equation",
"Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE",
"Linear Second Order ODE/y'' + y' - 12 y = 0",
"Definition:Differential Equation/Solution/General Solution"
] |
proofwiki-11167 | Cardinality of Image of Set not greater than Cardinality of Set | Let $X, Y$ be sets.
Let $f: X \to Y$ be a mapping.
Let $A$ be a subset of $X$.
Then:
:$\card {\map {f^\to} A} \le \card A$
where $\card A$ denotes the cardinality of $A$. | By definitions of surjection and restriction of mapping:
:$F \restriction_A: A \to \map {f^\to} A$ is a surjection
Thus by Surjection iff Cardinal Inequality:
:$\card {\map {f^\to} A} \le \card A$
{{qed}} | Let $X, Y$ be [[Definition:Set|sets]].
Let $f: X \to Y$ be a [[Definition:Mapping|mapping]].
Let $A$ be a [[Definition:Subset|subset]] of $X$.
Then:
:$\card {\map {f^\to} A} \le \card A$
where $\card A$ denotes the [[Definition:Cardinality|cardinality]] of $A$. | By definitions of [[Definition:Surjection|surjection]] and [[Definition:Restriction of Mapping|restriction of mapping]]:
:$F \restriction_A: A \to \map {f^\to} A$ is a [[Definition:Surjection|surjection]]
Thus by [[Surjection iff Cardinal Inequality]]:
:$\card {\map {f^\to} A} \le \card A$
{{qed}} | Cardinality of Image of Set not greater than Cardinality of Set | https://proofwiki.org/wiki/Cardinality_of_Image_of_Set_not_greater_than_Cardinality_of_Set | https://proofwiki.org/wiki/Cardinality_of_Image_of_Set_not_greater_than_Cardinality_of_Set | [
"Cardinals",
"Images"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Subset",
"Definition:Cardinality"
] | [
"Definition:Surjection",
"Definition:Restriction/Mapping",
"Definition:Surjection",
"Surjection iff Cardinal Inequality"
] |
proofwiki-11168 | Projector has Norm 1 | An idempotent operator $P$ is a projector on the Hilbert Space $H$ {{iff}} $P$ has norm $1$:
:$\ds \norm P \equiv \sup_{x \mathop \in H} \frac {\norm P} {\norm x} = 1$ | For all $x \in \Rng P$:
:$\norm {P \dfrac x {\norm x} } = \dfrac {\norm x} {\norm x} = 1$
so $\norm P \ge 1$.
It remains to show that this holds with equality {{iff}} $P$ is a projector.
First, suppose $P$ is a projector.
Let $\set {p_1, p_2, \ldots}$ be an orthonormal basis for $\Rng P$.
Let $\set {q_1, q_2, \ldots}$ ... | An [[Idempotent Operators|idempotent operator]] $P$ is a [[projector]] on the [[Definition:Hilbert Space|Hilbert Space]] $H$ {{iff}} $P$ has [[Definition:Norm on Bounded Linear Transformation|norm]] $1$:
:$\ds \norm P \equiv \sup_{x \mathop \in H} \frac {\norm P} {\norm x} = 1$ | For all $x \in \Rng P$:
:$\norm {P \dfrac x {\norm x} } = \dfrac {\norm x} {\norm x} = 1$
so $\norm P \ge 1$.
It remains to show that this holds with equality {{iff}} $P$ is a projector.
First, suppose $P$ is a projector.
Let $\set {p_1, p_2, \ldots}$ be an [[Definition:Orthonormal Subset|orthonormal]] [[Definition:... | Projector has Norm 1 | https://proofwiki.org/wiki/Projector_has_Norm_1 | https://proofwiki.org/wiki/Projector_has_Norm_1 | [
"Hilbert Spaces"
] | [
"Idempotent Operators",
"projector",
"Definition:Hilbert Space",
"Definition:Norm/Bounded Linear Transformation"
] | [
"Definition:Orthonormal Subset",
"Definition:Basis (Linear Algebra)",
"Definition:Orthonormal Subset",
"Definition:Basis (Linear Algebra)",
"Pythagoras's Theorem (Inner Product Space)",
"Pythagoras's Theorem (Inner Product Space)",
"Pythagoras's Theorem (Inner Product Space)",
"Idempotent Operators",
... |
proofwiki-11169 | Linear Second Order ODE/y'' + 3 y' - 10 y = 0 | The second order ODE:
:$(1): \quad y' ' + 3 y' - 10 y = 0$
has the general solution:
:$y = C_1 e^{2 x} + C_2 e^{-5 x}$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad: m^2 + 3 m - 10 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = 2$
:$m_2 = -5$
These are real and unequal.
So from Solution of Constant Coeffi... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' + 3 y' - 10 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^{2 x} + C_2 e^{-5 x}$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad: m^2 + 3 m - 10 = 0$
From [[Solution to Quadratic Equation with Real Coefficients]]... | Linear Second Order ODE/y'' + 3 y' - 10 y = 0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_3_y'_-_10_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_3_y'_-_10_y_=_0 | [
"Examples of Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Definition:Real Number",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential Equ... |
proofwiki-11170 | Linear Second Order ODE/y'' + 3 y' - 10 y = 6 exp 4 x | The second order ODE:
:$(1): \quad y' ' + 3 y' - 10 y = 6 e^{4 x}$
has the general solution:
:$y = C_1 e^{2 x} + C_2 e^{-5 x} + \dfrac {e^{4 x} } 3$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
:$y' ' + p y' + q y = \map R x$
where:
:$p = 3$
:$q = -10$
:$\map R x = 6 e^{4 x}$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$y' ' + 3 y' - ... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' + 3 y' - 10 y = 6 e^{4 x}$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^{2 x} + C_2 e^{-5 x} + \dfrac {e^{4 x} } 3$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] with [[Definition:Constant|constant]] [[Definition:Coefficient|coefficients]] in the form:
:$y' ' + p y' + q y = \map R x$
where:
:$p = 3$
:$q = -10$
:$\map R x = 6 e^{4 x}$
First we establish t... | Linear Second Order ODE/y'' + 3 y' - 10 y = 6 exp 4 x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_3_y'_-_10_y_=_6_exp_4_x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_3_y'_-_10_y_=_6_exp_4_x | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Undetermined Coefficients"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Constant",
"Definition:Coefficient",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' + 3 y' - 10 y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Method of Und... |
proofwiki-11171 | Linear Second Order ODE/y'' + 4 y = 0 | The second order ODE:
:$(1): \quad y' ' + 4 y = 0$
has the general solution:
:$y = C_1 \cos 2 x + C_2 \sin 2 x$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad: m^2 + 4 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = 2 i$
:$m_2 = -2 i$
These are complex and unequal.
So from Solution of Constant Coeffi... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' + 4 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 \cos 2 x + C_2 \sin 2 x$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad: m^2 + 4 = 0$
From [[Solution to Quadratic Equation with Real Coefficients]], the [... | Linear Second Order ODE/y'' + 4 y = 0/Proof 1 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_4_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_4_y_=_0/Proof_1 | [
"Linear Second Order ODE/y'' + 4 y = 0",
"Examples of Linear Second Order ODE/y'' + k^2 y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Definition:Complex Number",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential ... |
proofwiki-11172 | Linear Second Order ODE/y'' + 4 y = 0 | The second order ODE:
:$(1): \quad y' ' + 4 y = 0$
has the general solution:
:$y = C_1 \cos 2 x + C_2 \sin 2 x$ | This is an instance of:
:Linear Second Order ODE: $y' ' + k^2 y = 0$
which yields:
:$y = C_1 \cos k x + C_2 \sin k x$
where $k = 2$.
Hence the result.
{{qed}} | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' + 4 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 \cos 2 x + C_2 \sin 2 x$ | This is an instance of:
:[[Linear Second Order ODE/y'' + k^2 y = 0|Linear Second Order ODE: $y' ' + k^2 y = 0$]]
which yields:
:$y = C_1 \cos k x + C_2 \sin k x$
where $k = 2$.
Hence the result.
{{qed}} | Linear Second Order ODE/y'' + 4 y = 0/Proof 2 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_4_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_4_y_=_0/Proof_2 | [
"Linear Second Order ODE/y'' + 4 y = 0",
"Examples of Linear Second Order ODE/y'' + k^2 y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Linear Second Order ODE/y'' + k^2 y = 0"
] |
proofwiki-11173 | Linear Second Order ODE/y'' + 4 y = 3 sine x | The second order ODE:
:$(1): \quad y' ' + 4 y = 3 \sin x$
has the general solution:
:$y = C_1 \cos 2 x + C_2 \sin 2 x + \sin x$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
:$y' ' + p y' + q y = \map R x$
where:
:$p = 0$
:$q = 4$
:$\map R x = 3 \sin x$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$(2): \quad y' ' +... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' + 4 y = 3 \sin x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 \cos 2 x + C_2 \sin 2 x + \sin x$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] with [[Definition:Constant|constant]] [[Definition:Coefficient|coefficients]] in the form:
:$y' ' + p y' + q y = \map R x$
where:
:$p = 0$
:$q = 4$
:$\map R x = 3 \sin x$
First we establish the ... | Linear Second Order ODE/y'' + 4 y = 3 sine x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_4_y_=_3_sine_x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_4_y_=_3_sine_x | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Undetermined Coefficients"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Constant",
"Definition:Coefficient",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' + 4 y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Differenti... |
proofwiki-11174 | Linear Second Order ODE/y'' + 10 y' + 25 y = 0 | The second order ODE:
:$(1): \quad y' ' + 10 y' + 25 y = 0$
has the general solution:
:$y = C_1 e^{-5 x} + C_2 x e^{-5 x}$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad: m^2 + 10 m + 25 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = m_2 = -5$
These are real and equal.
So from Solution of Constant Coefficient ... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' + 10 y' + 25 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^{-5 x} + C_2 x e^{-5 x}$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad: m^2 + 10 m + 25 = 0$
From [[Solution to Quadratic Equation with Real Coefficients]... | Linear Second Order ODE/y'' + 10 y' + 25 y = 0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_10_y'_+_25_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_10_y'_+_25_y_=_0 | [
"Examples of Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Definition:Real Number",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential Equ... |
proofwiki-11175 | Linear Second Order ODE/y'' + 10 y' + 25 y = 14 exp -5 x | The second order ODE:
:$(1): \quad y'' + 10 y' + 25 y = 14 e^{-5 x}$
has the general solution:
:$y = C_1 \cos 2 x + C_2 \sin 2 x + \sin x$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 10$
:$q = 25$
:$\map R x = 14 e^{-5 x}$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$(2): \quad y'... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' + 10 y' + 25 y = 14 e^{-5 x}$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 \cos 2 x + C_2 \sin 2 x + \sin x$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] with [[Definition:Constant|constant]] [[Definition:Coefficient|coefficients]] in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 10$
:$q = 25$
:$\map R x = 14 e^{-5 x}$
First we establish ... | Linear Second Order ODE/y'' + 10 y' + 25 y = 14 exp -5 x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_10_y'_+_25_y_=_14_exp_-5_x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_10_y'_+_25_y_=_14_exp_-5_x | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Undetermined Coefficients"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Constant",
"Definition:Coefficient",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' + 10 y' + 25 y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:D... |
proofwiki-11176 | Linear Second Order ODE/y'' - 2 y' + 5 y = 0 | The second order ODE:
:$(1): \quad y' ' - 2 y' + 5 y = 0$
has the general solution:
:$y = e^x \paren {C_1 \cos 2 x + C_2 \sin 2 x}$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad: m^2 - 2 m + 5 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = 1 + 2 i$
:$m_2 = 1 - 2 i$
So from Solution of Constant Coefficient Homogeneous ... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' - 2 y' + 5 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = e^x \paren {C_1 \cos 2 x + C_2 \sin 2 x}$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad: m^2 - 2 m + 5 = 0$
From [[Solution to Quadratic Equation with Real Coefficients]]... | Linear Second Order ODE/y'' - 2 y' + 5 y = 0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_+_5_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_+_5_y_=_0 | [
"Examples of Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential Equation/Solution/General Solut... |
proofwiki-11177 | Linear Second Order ODE/y'' - 2 y' + 5 y = 25 x^2 + 12 | The second order ODE:
:$(1): \quad y'' - 2 y' + 5 y = 25 x^2 + 12$
has the general solution:
:$y = e^x \paren {C_1 \cos 2 x + C_2 \sin 2 x} + 2 + 4 x + 5 x^2$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = -2$
:$q = 5$
:$\map R x = 25 x^2 + 12$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$(2): \quad y''... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' - 2 y' + 5 y = 25 x^2 + 12$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = e^x \paren {C_1 \cos 2 x + C_2 \sin 2 x} + 2 + 4 x + 5 x^2$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] with [[Definition:Constant|constant]] [[Definition:Coefficient|coefficients]] in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = -2$
:$q = 5$
:$\map R x = 25 x^2 + 12$
First we establish t... | Linear Second Order ODE/y'' - 2 y' + 5 y = 25 x^2 + 12 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_+_5_y_=_25_x^2_+_12 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_+_5_y_=_25_x^2_+_12 | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Undetermined Coefficients"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Constant",
"Definition:Coefficient",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' - 2 y' + 5 y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Dif... |
proofwiki-11178 | Linear Second Order ODE/y'' + y' - 12 y = 0 | The second order ODE:
:$(1): \quad y' ' + y' - 12 y = 0$
has the general solution:
:$y = C_1 e^{3 x} + C_2 e^{-4 x}$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad: m^2 + m - 12 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = 3$
:$m_2 = -4$
These are real and unequal.
So from Solution of Constant Coeffici... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' + y' - 12 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^{3 x} + C_2 e^{-4 x}$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad: m^2 + m - 12 = 0$
From [[Solution to Quadratic Equation with Real Coefficients]], ... | Linear Second Order ODE/y'' + y' - 12 y = 0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_y'_-_12_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_y'_-_12_y_=_0 | [
"Examples of Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Definition:Real Number",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential Equ... |
proofwiki-11179 | Linear Second Order ODE/y'' + 2 y' + 10 y = 0 | The second order ODE:
:$(1): \quad y' ' + 2 y' + 10 y = 0$
has the general solution:
:$y = e^{-x} \paren {C_1 \cos 3 x + C_2 \sin 3 x}$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad: m^2 + 2 m + 10 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = -1 + 3 i$
:$m_2 = -1 - 3 i$
So from Solution of Constant Coefficient Homogeneo... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' + 2 y' + 10 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = e^{-x} \paren {C_1 \cos 3 x + C_2 \sin 3 x}$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad: m^2 + 2 m + 10 = 0$
From [[Solution to Quadratic Equation with Real Coefficients]]... | Linear Second Order ODE/y'' + 2 y' + 10 y = 0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_2_y'_+_10_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_2_y'_+_10_y_=_0 | [
"Examples of Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential Equation/Solution/General Solut... |
proofwiki-11180 | Linear Second Order ODE/y'' + 4 y' + 4 y = 0 | The second order ODE:
:$(1): \quad y' ' + 4 y' + 4 y = 0$
has the general solution:
:$y = \paren {C_1 + C_2 x} e^{-2 x}$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad: m^2 + 4 m + 4 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = m_2 = -2$
These are real and equal.
So from Solution of Constant Coefficient Ho... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' + 4 y' + 4 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = \paren {C_1 + C_2 x} e^{-2 x}$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad: m^2 + 4 m + 4 = 0$
From [[Solution to Quadratic Equation with Real Coefficients]],... | Linear Second Order ODE/y'' + 4 y' + 4 y = 0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_4_y'_+_4_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_4_y'_+_4_y_=_0 | [
"Examples of Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Definition:Real Number",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential Equ... |
proofwiki-11181 | Linear Second Order ODE/y'' + y' - 2 y = 0 | The second order ODE:
:$(1): \quad y' ' + y' - 2 y = 0$
has the general solution:
:$y = C_1 e^x + C_2 e^{-2 x}$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad: m^2 + m - 2 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = 1$
:$m_2 = -2$
These are real and unequal.
So from Solution of Constant Coefficie... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' + y' - 2 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^x + C_2 e^{-2 x}$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad: m^2 + m - 2 = 0$
From [[Solution to Quadratic Equation with Real Coefficients]], t... | Linear Second Order ODE/y'' + y' - 2 y = 0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_y'_-_2_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_y'_-_2_y_=_0 | [
"Examples of Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Definition:Real Number",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential Equ... |
proofwiki-11182 | Linear Second Order ODE/y'' - 4 y = 0 | The second order ODE:
:$(1): \quad y' ' - 4 y = 0$
has the general solution:
:$y = C_1 e^{2 x} + C_2 e^{-2 x}$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad: m^2 - 4 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = 2$
:$m_2 = -2$
These are complex and unequal.
So from Solution of Constant Coefficien... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' - 4 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^{2 x} + C_2 e^{-2 x}$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad: m^2 - 4 = 0$
From [[Solution to Quadratic Equation with Real Coefficients]], the [... | Linear Second Order ODE/y'' - 4 y = 0/Proof 1 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_4_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_4_y_=_0/Proof_1 | [
"Linear Second Order ODE/y'' - 4 y = 0",
"Examples of Linear Second Order ODE/y'' - k^2 y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Definition:Real Number",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential Equ... |
proofwiki-11183 | Linear Second Order ODE/y'' - 4 y = 0 | The second order ODE:
:$(1): \quad y' ' - 4 y = 0$
has the general solution:
:$y = C_1 e^{2 x} + C_2 e^{-2 x}$ | This is an instance of:
:Linear Second Order ODE: $y'' - k^2 y = 0$
which yields:
:$y = C_1 e^{k x} + C_2 e^{-k x}$
where $k = 2$.
Hence the result.
{{qed}} | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' - 4 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^{2 x} + C_2 e^{-2 x}$ | This is an instance of:
:[[Linear Second Order ODE/y'' - k^2 y = 0|Linear Second Order ODE: $y'' - k^2 y = 0$]]
which yields:
:$y = C_1 e^{k x} + C_2 e^{-k x}$
where $k = 2$.
Hence the result.
{{qed}} | Linear Second Order ODE/y'' - 4 y = 0/Proof 2 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_4_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_4_y_=_0/Proof_2 | [
"Linear Second Order ODE/y'' - 4 y = 0",
"Examples of Linear Second Order ODE/y'' - k^2 y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Linear Second Order ODE/y'' - k^2 y = 0"
] |
proofwiki-11184 | Linear Second Order ODE/y'' - y' - 6 y = 0 | The second order ODE:
:$(1): \quad y' ' - y' - 6 y = 0$
has the general solution:
:$y = C_1 e^{3 x} + C_2 e^{-2 x}$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad: m^2 - m - 6 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = 3$
:$m_2 = -2$
These are real and unequal.
So from Solution of Constant Coefficie... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' - y' - 6 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^{3 x} + C_2 e^{-2 x}$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad: m^2 - m - 6 = 0$
From [[Solution to Quadratic Equation with Real Coefficients]], t... | Linear Second Order ODE/y'' - y' - 6 y = 0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_y'_-_6_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_y'_-_6_y_=_0 | [
"Examples of Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Definition:Real Number",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential Equ... |
proofwiki-11185 | Linear Second Order ODE/y'' - y' - 6 y = 20 exp -2 x | The second order ODE:
:$(1): \quad y'' - y' - 6 y = 20 e^{-2 x}$
has the general solution:
:$y = C_1 e^{3 x} + C_2 e^{-2 x} - 4 x e^{-2 x}$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = -1$
:$q = -6$
:$\map R x = 20 e^{-2 x}$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$y'' - y' - 6 ... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' - y' - 6 y = 20 e^{-2 x}$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^{3 x} + C_2 e^{-2 x} - 4 x e^{-2 x}$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] with [[Definition:Constant|constant]] [[Definition:Coefficient|coefficients]] in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = -1$
:$q = -6$
:$\map R x = 20 e^{-2 x}$
First we establish ... | Linear Second Order ODE/y'' - y' - 6 y = 20 exp -2 x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_y'_-_6_y_=_20_exp_-2_x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_y'_-_6_y_=_20_exp_-2_x | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Undetermined Coefficients"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Constant",
"Definition:Coefficient",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' - y' - 6 y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Method of Undete... |
proofwiki-11186 | Linear Second Order ODE/y'' - 3 y' + 2 y = 14 sine 2 x - 18 cosine 2 x | The second order ODE:
:$(1): \quad y'' - 3 y' + 2 y = 14 \sin 2 x - 18 \cos 2 x$
has the general solution:
:$y = C_1 e^{3 x} + C_2 e^{-2 x} - 4 x e^{-2 x}$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = -3$
:$q = 2$
:$\map R x = 14 \sin 2 x - 18 \cos 2 x$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' - 3 y' + 2 y = 14 \sin 2 x - 18 \cos 2 x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^{3 x} + C_2 e^{-2 x} - 4 x e^{-2 x}$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] with [[Definition:Constant|constant]] [[Definition:Coefficient|coefficients]] in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = -3$
:$q = 2$
:$\map R x = 14 \sin 2 x - 18 \cos 2 x$
First ... | Linear Second Order ODE/y'' - 3 y' + 2 y = 14 sine 2 x - 18 cosine 2 x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_3_y'_+_2_y_=_14_sine_2_x_-_18_cosine_2_x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_3_y'_+_2_y_=_14_sine_2_x_-_18_cosine_2_x | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Undetermined Coefficients"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Constant",
"Definition:Coefficient",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' - 3 y' + 2 y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Dif... |
proofwiki-11187 | Linear Second Order ODE/y'' + y = 2 cosine x | The second order ODE:
:$(1): \quad y'' + y = 2 \cos x$
has the general solution:
:$y = C_1 \sin x + C_2 \cos x + x \cos x$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 0$
:$q = 1$
:$\map R x = 2 \cos x$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$(2): \quad y'' + y... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' + y = 2 \cos x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 \sin x + C_2 \cos x + x \cos x$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] with [[Definition:Constant|constant]] [[Definition:Coefficient|coefficients]] in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 0$
:$q = 1$
:$\map R x = 2 \cos x$
First we establish the s... | Linear Second Order ODE/y'' + y = 2 cosine x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_y_=_2_cosine_x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_y_=_2_cosine_x | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Undetermined Coefficients"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Constant",
"Definition:Coefficient",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' + y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Differential... |
proofwiki-11188 | Linear Second Order ODE/y'' - 2 y' = 0 | The second order ODE:
:$(1): \quad y' ' - 2 y' = 0$
has the general solution:
:$y = C_1 + C_2 e^{2 x}$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad: m^2 - 2 m = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = 0$
:$m_2 = 2$
These are real and unequal.
So from Solution of Constant Coefficient ... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' - 2 y' = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 + C_2 e^{2 x}$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad: m^2 - 2 m = 0$
From [[Solution to Quadratic Equation with Real Coefficients]], th... | Linear Second Order ODE/y'' - 2 y' = 0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_=_0 | [
"Examples of Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Definition:Real Number",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential Equ... |
proofwiki-11189 | Linear Second Order ODE/y'' - 2 y' = 12 x - 10 | The second order ODE:
:$(1): \quad y'' - 2 y' = 12 x - 10$
has the general solution:
:$y = C_1 + C_2 e^{2 x} + 2 x - 3 x^2$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 2$
:$q = 0$
:$\map R x = 12 x - 10$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$y'' - 2 y' = 0$
F... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' - 2 y' = 12 x - 10$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 + C_2 e^{2 x} + 2 x - 3 x^2$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] with [[Definition:Constant|constant]] [[Definition:Coefficient|coefficients]] in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 2$
:$q = 0$
:$\map R x = 12 x - 10$
First we establish the ... | Linear Second Order ODE/y'' - 2 y' = 12 x - 10 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_=_12_x_-_10 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_=_12_x_-_10 | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Undetermined Coefficients"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Constant",
"Definition:Coefficient",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' - 2 y' = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Different... |
proofwiki-11190 | Directed iff Finite Subsets have Upper Bounds | Let $\struct {S, \precsim}$ be a preordered set.
Let $H$ be a non-empty subset of $S$.
Then $H$ is directed {{iff}}:
:for every finite subset $A$ of $H$:
::$\exists h \in H: \forall a \in A: a \precsim h$ | === Sufficient Condition ===
Let $R$ be directed.
We will prove by induction of the cardinality of finite subset of $H$. | Let $\struct {S, \precsim}$ be a [[Definition:Preordered Set|preordered set]].
Let $H$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $S$.
Then $H$ is [[Definition:Directed Subset|directed]] {{iff}}:
:for every [[Definition:Finite Set|finite]] [[Definition:Subset|subset]] $A$ of $H$:
::$... | === Sufficient Condition ===
Let $R$ be [[Definition:Directed Subset|directed]].
We will prove by [[Principle of Mathematical Induction|induction]] of the [[Definition:Cardinality|cardinality]] of [[Definition:Finite Set|finite]] [[Definition:Subset|subset]] of $H$. | Directed iff Finite Subsets have Upper Bounds | https://proofwiki.org/wiki/Directed_iff_Finite_Subsets_have_Upper_Bounds | https://proofwiki.org/wiki/Directed_iff_Finite_Subsets_have_Upper_Bounds | [
"Preorder Theory"
] | [
"Definition:Preordering/Preordered Set",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Directed Subset",
"Definition:Finite Set",
"Definition:Subset"
] | [
"Definition:Directed Subset",
"Principle of Mathematical Induction",
"Definition:Cardinality",
"Definition:Finite Set",
"Definition:Subset",
"Definition:Cardinality",
"Definition:Subset",
"Definition:Subset",
"Definition:Directed Subset",
"Definition:Finite Set",
"Definition:Subset",
"Definiti... |
proofwiki-11191 | Linear Second Order ODE/y'' + k^2 y = sine b x | The second order ODE:
:$(1): \quad y'' + k^2 y = \sin b x$
has the general solution:
:$y = \begin{cases} C_1 \sin k x + C_2 \cos k x + \dfrac {\sin b x} {k^2 - b^2} & : b \ne k \\
C_1 \sin k x + C_2 \cos k x - \dfrac {x \cos k x} {2 k} & : b = k
\end{cases}$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 0$
:$q = k^2$
:$\map R x = \sin b x$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$(2): \quad y'' +... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' + k^2 y = \sin b x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = \begin{cases} C_1 \sin k x + C_2 \cos k x + \dfrac {\sin b x} {k^2 - b^2} & : b \ne k \\
C_1 \sin k x + C_2 \cos k x - \dfrac {x \cos k x} ... | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] with [[Definition:Constant|constant]] [[Definition:Coefficient|coefficients]] in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 0$
:$q = k^2$
:$\map R x = \sin b x$
First we establish the... | Linear Second Order ODE/y'' + k^2 y = sine b x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_k^2_y_=_sine_b_x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_k^2_y_=_sine_b_x | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Undetermined Coefficients"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Constant",
"Definition:Coefficient",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' + k^2 y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Differen... |
proofwiki-11192 | Combination of Solutions to Non-Homogeneous LSOODE with same Homogeneous Part | Let $\map {y_1} x$ be a particular solution of the linear second order ODE:
:$(1): \quad y' ' + \map P x y' + \map Q x y = \map {R_1} x$
Let $\map {y_2} x$ be a particular solution of the linear second order ODE:
:$(2): \quad y' ' + \map P x y' + \map Q x y = \map {R_2} x$
Then $\map y x = \map {y_1} x + \map {y_2} x$ ... | {{begin-eqn}}
{{eqn | n = 4
| l = {y_1}' ' + \map P x {y_1}' + \map Q x y_1
| r = \map {R_1} x
| c = as $y_1$ is a particular solution to $(1)$
}}
{{eqn | n = 5
| l = {y_2}' ' + \map P x {y_2}' + \map Q x y_2
| r = \map {R_2} x
| c = as $y_2$ is a particular solution to $(2)$
}}
{{eq... | Let $\map {y_1} x$ be a [[Definition:Particular Solution to Differential Equation|particular solution]] of the [[Definition:Linear Second Order ODE|linear second order ODE]]:
:$(1): \quad y' ' + \map P x y' + \map Q x y = \map {R_1} x$
Let $\map {y_2} x$ be a [[Definition:Particular Solution to Differential Equation|p... | {{begin-eqn}}
{{eqn | n = 4
| l = {y_1}' ' + \map P x {y_1}' + \map Q x y_1
| r = \map {R_1} x
| c = as $y_1$ is a [[Definition:Particular Solution to Differential Equation|particular solution]] to $(1)$
}}
{{eqn | n = 5
| l = {y_2}' ' + \map P x {y_2}' + \map Q x y_2
| r = \map {R_2} x
... | Combination of Solutions to Non-Homogeneous LSOODE with same Homogeneous Part | https://proofwiki.org/wiki/Combination_of_Solutions_to_Non-Homogeneous_LSOODE_with_same_Homogeneous_Part | https://proofwiki.org/wiki/Combination_of_Solutions_to_Non-Homogeneous_LSOODE_with_same_Homogeneous_Part | [
"Linear Second Order ODEs"
] | [
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Linear Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Linear Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/Parti... | [
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Differential Equation/Solution/Particular Solution",
"Linear Combination of Derivatives"
] |
proofwiki-11193 | Linear Second Order ODE/y'' + 4 y = 4 cosine 2 x + 6 cosine x + 8 x^2 - 4 x | The second order ODE:
:$(1): \quad y'' + 4 y = 4 \cos 2 x + 6 \cos x + 8 x^2 - 4 x$
has the general solution:
:$y = C_1 \sin 2 x + C_2 \cos 2 x + x \sin 2 x + 2 \cos x - 1 - x + 2 x^2$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 0$
:$q = 4$
:$\map R x = 4 \cos 2 x + 6 \cos x + 8 x^2 - 4 x$
First we establish the solution of the corresponding constant coefficient homogeneous linear second orde... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' + 4 y = 4 \cos 2 x + 6 \cos x + 8 x^2 - 4 x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 \sin 2 x + C_2 \cos 2 x + x \sin 2 x + 2 \cos x - 1 - x + 2 x^2$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] with [[Definition:Constant|constant]] [[Definition:Coefficient|coefficients]] in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 0$
:$q = 4$
:$\map R x = 4 \cos 2 x + 6 \cos x + 8 x^2 - 4 x$... | Linear Second Order ODE/y'' + 4 y = 4 cosine 2 x + 6 cosine x + 8 x^2 - 4 x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_4_y_=_4_cosine_2_x_+_6_cosine_x_+_8_x^2_-_4_x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_4_y_=_4_cosine_2_x_+_6_cosine_x_+_8_x^2_-_4_x | [
"Examples of Constant Coefficient LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Constant",
"Definition:Coefficient",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' + 4 y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Linear Second Order O... |
proofwiki-11194 | Linear Second Order ODE/y'' + 4 y = 4 cosine 2 x | The second order ODE:
:$(1): \quad y' ' + 4 y = 4 \cos 2 x$
has the general solution:
:$y = C_1 \sin 2 x + C_2 \cos 2 x + x \sin 2 x$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
:$y' ' + p y' + q y = \map R x$
where:
:$p = 0$
:$q = 4$
:$\map R x = 4 \cos 2 x$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$(2): \quad y' '... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' + 4 y = 4 \cos 2 x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 \sin 2 x + C_2 \cos 2 x + x \sin 2 x$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] with [[Definition:Constant|constant]] [[Definition:Coefficient|coefficients]] in the form:
:$y' ' + p y' + q y = \map R x$
where:
:$p = 0$
:$q = 4$
:$\map R x = 4 \cos 2 x$
First we establish th... | Linear Second Order ODE/y'' + 4 y = 4 cosine 2 x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_4_y_=_4_cosine_2_x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_4_y_=_4_cosine_2_x | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Undetermined Coefficients"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Constant",
"Definition:Coefficient",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' + 4 y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Differenti... |
proofwiki-11195 | Linear Second Order ODE/y'' + 4 y = 6 cosine x | The second order ODE:
:$(1): \quad y'' + 4 y = 6 \cos x$
has the general solution:
:$y = C_1 \sin 2 x + C_2 \cos 2 x + 2 \cos x$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 0$
:$q = 4$
:$\map R x = 6 \cos x$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$(2): \quad y'' + 4... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' + 4 y = 6 \cos x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 \sin 2 x + C_2 \cos 2 x + 2 \cos x$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] with [[Definition:Constant|constant]] [[Definition:Coefficient|coefficients]] in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 0$
:$q = 4$
:$\map R x = 6 \cos x$
First we establish the s... | Linear Second Order ODE/y'' + 4 y = 6 cosine x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_4_y_=_6_cosine_x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_4_y_=_6_cosine_x | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Undetermined Coefficients"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Constant",
"Definition:Coefficient",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' + 4 y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Differenti... |
proofwiki-11196 | Linear Second Order ODE/y'' + 4 y = 8 x^2 - 4 x | The second order ODE:
:$(1): \quad y'' + 4 y = 8 x^2 - 4 x$
has the general solution:
:$y = C_1 \sin 2 x + C_2 \cos 2 x - 1 - x + 2 x^2$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 0$
:$q = 4$
:$\map R x = 8 x^2 - 4 x$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$(2): \quad y'' ... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' + 4 y = 8 x^2 - 4 x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 \sin 2 x + C_2 \cos 2 x - 1 - x + 2 x^2$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] with [[Definition:Constant|constant]] [[Definition:Coefficient|coefficients]] in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 0$
:$q = 4$
:$\map R x = 8 x^2 - 4 x$
First we establish th... | Linear Second Order ODE/y'' + 4 y = 8 x^2 - 4 x/Proof 1 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_4_y_=_8_x^2_-_4_x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_4_y_=_8_x^2_-_4_x/Proof_1 | [
"Examples of Constant Coefficient LSOODEs",
"Linear Second Order ODE/y'' + 4 y = 8 x^2 - 4 x"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Constant",
"Definition:Coefficient",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' + 4 y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Method of Undetermine... |
proofwiki-11197 | Linear Second Order ODE/y'' + 4 y = 8 x^2 - 4 x | The second order ODE:
:$(1): \quad y'' + 4 y = 8 x^2 - 4 x$
has the general solution:
:$y = C_1 \sin 2 x + C_2 \cos 2 x - 1 - x + 2 x^2$ | Taking Laplace transforms:
:$\laptrans {y'' + 4 y} = \laptrans {8 x^2 - 4 x}$
We have:
{{begin-eqn}}
{{eqn | l = \laptrans {y'' + 4 y}
| r = \laptrans {y''} + 4 \laptrans y
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = s^2 \laptrans y - s \map y 0 - \map {y'} 0
| c = Laplace Transform of Second Der... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' + 4 y = 8 x^2 - 4 x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 \sin 2 x + C_2 \cos 2 x - 1 - x + 2 x^2$ | Taking [[Definition:Laplace Transform|Laplace transforms]]:
:$\laptrans {y'' + 4 y} = \laptrans {8 x^2 - 4 x}$
We have:
{{begin-eqn}}
{{eqn | l = \laptrans {y'' + 4 y}
| r = \laptrans {y''} + 4 \laptrans y
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = s^2 \laptrans y - s \map y 0 - \map {y'} ... | Linear Second Order ODE/y'' + 4 y = 8 x^2 - 4 x/Proof 2 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_4_y_=_8_x^2_-_4_x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_4_y_=_8_x^2_-_4_x/Proof_2 | [
"Examples of Constant Coefficient LSOODEs",
"Linear Second Order ODE/y'' + 4 y = 8 x^2 - 4 x"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Laplace Transform",
"Linear Combination of Laplace Transforms",
"Laplace Transform of Second Derivative",
"Linear Combination of Laplace Transforms",
"Laplace Transform of Power",
"Linear Combination of Laplace Transforms",
"Laplace Transform of Cosine",
"Laplace Transform of Sine",
"Lap... |
proofwiki-11198 | Linear Second Order ODE/y'' + y = cosecant x | The second order ODE:
:$(1): \quad y'' + y = \csc x$
has the general solution:
:$y = C_1 \sin x + C_2 \cos x - x \cos x + \sin x \map \ln {\sin x}$ | It can be seen that $\paren 1$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 0$
:$q = 1$
:$\map R x = \csc x$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$\paren 2: \quad... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' + y = \csc x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 \sin x + C_2 \cos x - x \cos x + \sin x \map \ln {\sin x}$ | It can be seen that $\paren 1$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] with [[Definition:Constant|constant]] [[Definition:Coefficient|coefficients]] in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 0$
:$q = 1$
:$\map R x = \csc x$
First we establish th... | Linear Second Order ODE/y'' + y = cosecant x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_y_=_cosecant_x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_y_=_cosecant_x | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Variation of Parameters"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Constant",
"Definition:Coefficient",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' + y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Differential... |
proofwiki-11199 | Linear Second Order ODE/y'' - 2 y' + y = 0 | The second order ODE:
:$(1): \quad y' ' - 2 y' + y = 0$
has the general solution:
:$y = C_1 e^x + C_2 x e^x$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad: m^2 - 2 m + 1 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = m_2 = 1$
These are real and equal.
So from Solution of Constant Coefficient Hom... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' - 2 y' + y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^x + C_2 x e^x$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad: m^2 - 2 m + 1 = 0$
From [[Solution to Quadratic Equation with Real Coefficients]],... | Linear Second Order ODE/y'' - 2 y' + y = 0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_+_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_+_y_=_0 | [
"Examples of Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Definition:Real Number",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential Equ... |
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