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 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-10900 | Orthogonal Trajectories/Examples/Rectangular Hyperbolas | Consider the one-parameter family of curves of rectangular hyperbolas:
:$(1): \quad x y = c$
Its family of orthogonal trajectories is given by the equation:
:$x^2 - y^2 = c$
:600px | We use the technique of formation of ordinary differential equation by elimination.
Differentiating $(1)$ {{WRT|Differentiation}} $x$ gives:
:$x \dfrac {\d y} {\d x} + y = 0$
{{begin-eqn}}
{{eqn | l = x \frac {\d y} {\d x} + y
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d y} {\d x}
| r ... | Consider the [[Definition:One-Parameter Family of Curves|one-parameter family of curves]] of [[Definition:Rectangular Hyperbola|rectangular hyperbolas]]:
:$(1): \quad x y = c$
Its [[Definition:Orthogonal Trajectories|family of orthogonal trajectories]] is given by the equation:
:$x^2 - y^2 = c$
:[[File:RectanguleHy... | We use the technique of [[Definition:Formation of Ordinary Differential Equation by Elimination|formation of ordinary differential equation by elimination]].
[[Definition:Differentiation|Differentiating]] $(1)$ {{WRT|Differentiation}} $x$ gives:
:$x \dfrac {\d y} {\d x} + y = 0$
{{begin-eqn}}
{{eqn | l = x \frac {\d ... | Orthogonal Trajectories/Examples/Rectangular Hyperbolas | https://proofwiki.org/wiki/Orthogonal_Trajectories/Examples/Rectangular_Hyperbolas | https://proofwiki.org/wiki/Orthogonal_Trajectories/Examples/Rectangular_Hyperbolas | [
"Examples of Orthogonal Trajectories",
"Hyperbolas"
] | [
"Definition:Family of Curves/One-Parameter",
"Definition:Rectangular Hyperbola",
"Definition:Orthogonal Trajectories",
"File:RectanguleHyperbolaeOrthogonalTrajectories.png"
] | [
"Definition:Formation of Ordinary Differential Equation by Elimination",
"Definition:Differentiation",
"Orthogonal Trajectories of One-Parameter Family of Curves",
"Definition:Orthogonal Trajectories",
"Solution to Separable Differential Equation"
] |
proofwiki-10901 | Orthogonal Trajectories/Examples/Parabolas Tangent to X Axis | Consider the one-parameter family of curves of parabolas which are tangent to the $x$-axis at the origin:
:$(1): \quad y = c x^2$
Its family of orthogonal trajectories is given by the equation:
:$x^2 + 2 y^2 = c$
:600px | We use the technique of formation of ordinary differential equation by elimination.
Differentiating $(1)$ {{WRT|Differentiation}} $x$ gives:
:$x \dfrac {\d y} {\d x} + y = 0$
{{begin-eqn}}
{{eqn | n = 2
| l = \frac {\d y} {\d x}
| r = 2 c x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d y} {\d x... | Consider the [[Definition:One-Parameter Family of Curves|one-parameter family of curves]] of [[Definition:Parabola|parabolas]] which are [[Definition:Tangent Line|tangent]] to the [[Definition:X-Axis|$x$-axis]] at the [[Definition:Origin|origin]]:
:$(1): \quad y = c x^2$
Its [[Definition:Orthogonal Trajectories|famil... | We use the technique of [[Definition:Formation of Ordinary Differential Equation by Elimination|formation of ordinary differential equation by elimination]].
[[Definition:Differentiation|Differentiating]] $(1)$ {{WRT|Differentiation}} $x$ gives:
:$x \dfrac {\d y} {\d x} + y = 0$
{{begin-eqn}}
{{eqn | n = 2
| l ... | Orthogonal Trajectories/Examples/Parabolas Tangent to X Axis | https://proofwiki.org/wiki/Orthogonal_Trajectories/Examples/Parabolas_Tangent_to_X_Axis | https://proofwiki.org/wiki/Orthogonal_Trajectories/Examples/Parabolas_Tangent_to_X_Axis | [
"Examples of Orthogonal Trajectories",
"Parabolas"
] | [
"Definition:Family of Curves/One-Parameter",
"Definition:Parabola",
"Definition:Tangent Line",
"Definition:Axis/X-Axis",
"Definition:Coordinate System/Origin",
"Definition:Orthogonal Trajectories",
"File:ParabolasTangentAxisOrthogonalTrajectories.png"
] | [
"Definition:Formation of Ordinary Differential Equation by Elimination",
"Definition:Differentiation",
"Orthogonal Trajectories of One-Parameter Family of Curves",
"Definition:Orthogonal Trajectories",
"Solution to Separable Differential Equation"
] |
proofwiki-10902 | Orthogonal Trajectories/Examples/Cardioids | Consider the one-parameter family of curves of cardioids given in polar form as:
:$(1): \quad r = c \paren {1 + \cos \theta}$
Its family of orthogonal trajectories is given by the equation:
:$r = c \paren {1 - \cos \theta}$
:600px | We use the technique of formation of ordinary differential equation by elimination.
Differentiating $(1)$ {{WRT|Differentiation}} $r$ gives:
{{begin-eqn}}
{{eqn | n = 2
| l = \frac {\d r} {\d \theta}
| r = - c \sin \theta
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d r} {\d \theta}
| r = ... | Consider the [[Definition:One-Parameter Family of Curves|one-parameter family of curves]] of [[Definition:Cardioid|cardioids]] given in [[Definition:Polar Coordinates|polar form]] as:
:$(1): \quad r = c \paren {1 + \cos \theta}$
Its [[Definition:Orthogonal Trajectories|family of orthogonal trajectories]] is given by ... | We use the technique of [[Definition:Formation of Ordinary Differential Equation by Elimination|formation of ordinary differential equation by elimination]].
[[Definition:Differentiation|Differentiating]] $(1)$ {{WRT|Differentiation}} $r$ gives:
{{begin-eqn}}
{{eqn | n = 2
| l = \frac {\d r} {\d \theta}
|... | Orthogonal Trajectories/Examples/Cardioids | https://proofwiki.org/wiki/Orthogonal_Trajectories/Examples/Cardioids | https://proofwiki.org/wiki/Orthogonal_Trajectories/Examples/Cardioids | [
"Examples of Orthogonal Trajectories",
"Cardioids"
] | [
"Definition:Family of Curves/One-Parameter",
"Definition:Cardioid",
"Definition:Polar Coordinates",
"Definition:Orthogonal Trajectories",
"File:CardioidsOrthogonalTrajectories.png"
] | [
"Definition:Formation of Ordinary Differential Equation by Elimination",
"Definition:Differentiation",
"Orthogonal Trajectories of One-Parameter Family of Curves",
"Definition:Orthogonal Trajectories",
"Solution to Separable Differential Equation"
] |
proofwiki-10903 | Orthogonal Trajectories/Examples/Exponential Functions | Consider the one-parameter family of curves of graphs of the exponential function:
:$(1): \quad y = c e^x$
Its family of orthogonal trajectories is given by the equation:
:$y^2 = -2 x + c$
:600px | We use the technique of formation of ordinary differential equation by elimination.
Differentiating $(1)$ {{WRT|Differentiation}} $x$ gives:
:$\dfrac {\d y} {\d x} = c e^x$
{{begin-eqn}}
{{eqn | n = 2
| l = \frac {\d y} {\d x}
| r = c e^x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d y} {\d x}
... | Consider the [[Definition:One-Parameter Family of Curves|one-parameter family of curves]] of [[Definition:Graph of Mapping|graphs]] of the [[Definition:Real Exponential Function|exponential function]]:
:$(1): \quad y = c e^x$
Its [[Definition:Orthogonal Trajectories|family of orthogonal trajectories]] is given by the... | We use the technique of [[Definition:Formation of Ordinary Differential Equation by Elimination|formation of ordinary differential equation by elimination]].
[[Definition:Differentiation|Differentiating]] $(1)$ {{WRT|Differentiation}} $x$ gives:
:$\dfrac {\d y} {\d x} = c e^x$
{{begin-eqn}}
{{eqn | n = 2
| l = ... | Orthogonal Trajectories/Examples/Exponential Functions | https://proofwiki.org/wiki/Orthogonal_Trajectories/Examples/Exponential_Functions | https://proofwiki.org/wiki/Orthogonal_Trajectories/Examples/Exponential_Functions | [
"Examples of Orthogonal Trajectories"
] | [
"Definition:Family of Curves/One-Parameter",
"Definition:Graph of Mapping",
"Definition:Exponential Function/Real",
"Definition:Orthogonal Trajectories",
"File:ExponentialsOrthogonalTrajectories.png"
] | [
"Definition:Formation of Ordinary Differential Equation by Elimination",
"Definition:Differentiation",
"Orthogonal Trajectories of One-Parameter Family of Curves",
"Definition:Orthogonal Trajectories",
"Solution to Separable Differential Equation"
] |
proofwiki-10904 | Orthogonal Trajectories/Examples/Parabolas with Focus at Origin | Consider the one-parameter family of curves of parabolas whose focus is at the origin and whose axis is the $x$-axis:
:$(1): \quad y^2 = 4 c \paren {x + c}$
Its family of orthogonal trajectories is given by the equation:
:$y^2 = 4 c \paren {x + c}$
:600px | We use the technique of formation of ordinary differential equation by elimination.
Differentiating $(1)$ {{WRT|Differentiation}} $x$ gives:
{{begin-eqn}}
{{eqn | n = 2
| l = 2 y \frac {\d y} {\d x}
| r = 4 c
| c =
}}
{{eqn | ll= \leadsto
| l = c
| r = \frac y 2 \frac {\d y} {\d x}
... | Consider the [[Definition:One-Parameter Family of Curves|one-parameter family of curves]] of [[Definition:Parabola|parabolas]] whose [[Definition:Focus of Parabola|focus]] is at the [[Definition:Origin|origin]] and whose [[Definition:Axis of Parabola|axis]] is the [[Definition:X-Axis|$x$-axis]]:
:$(1): \quad y^2 = 4 c ... | We use the technique of [[Definition:Formation of Ordinary Differential Equation by Elimination|formation of ordinary differential equation by elimination]].
[[Definition:Differentiation|Differentiating]] $(1)$ {{WRT|Differentiation}} $x$ gives:
{{begin-eqn}}
{{eqn | n = 2
| l = 2 y \frac {\d y} {\d x}
| ... | Orthogonal Trajectories/Examples/Parabolas with Focus at Origin | https://proofwiki.org/wiki/Orthogonal_Trajectories/Examples/Parabolas_with_Focus_at_Origin | https://proofwiki.org/wiki/Orthogonal_Trajectories/Examples/Parabolas_with_Focus_at_Origin | [
"Examples of Orthogonal Trajectories"
] | [
"Definition:Family of Curves/One-Parameter",
"Definition:Parabola",
"Definition:Parabola/Focus",
"Definition:Coordinate System/Origin",
"Definition:Parabola/Axis",
"Definition:Axis/X-Axis",
"Definition:Orthogonal Trajectories",
"File:ParabolasFocusOriginOrthogonalTrajectories.png"
] | [
"Definition:Formation of Ordinary Differential Equation by Elimination",
"Definition:Differentiation",
"Orthogonal Trajectories of One-Parameter Family of Curves",
"Definition:Orthogonal Trajectories"
] |
proofwiki-10905 | Half-Life of Radioactive Substance | Let a radioactive element $S$ decay with a rate constant $k$.
Then its half-life $T$ is given by:
:$T = \dfrac {\ln 2} k$
:400px | Let $x_0$ be the quantity of $S$ at time $t = 0$.
At time $t = T$ the quantity of $S$ has been reduced to $x = \dfrac {x_0} 2$.
This gives:
{{begin-eqn}}
{{eqn | l = x_0 e^{-k T}
| r = \frac {x_0} 2
| c = First-Order Reaction
}}
{{eqn | ll= \leadsto
| l = e^{k T}
| r = 2
| c =
}}
{{eqn | l... | Let a [[Definition:Radioactive Element|radioactive element]] $S$ [[Definition:Radioactive Decay|decay]] with a [[Definition:Rate Constant|rate constant]] $k$.
Then its [[Definition:Half-Life|half-life]] $T$ is given by:
:$T = \dfrac {\ln 2} k$
:[[File:HalfLife.png|400px]] | Let $x_0$ be the quantity of $S$ at [[Definition:Time|time]] $t = 0$.
At [[Definition:Time|time]] $t = T$ the quantity of $S$ has been reduced to $x = \dfrac {x_0} 2$.
This gives:
{{begin-eqn}}
{{eqn | l = x_0 e^{-k T}
| r = \frac {x_0} 2
| c = [[First-Order Reaction]]
}}
{{eqn | ll= \leadsto
| l =... | Half-Life of Radioactive Substance | https://proofwiki.org/wiki/Half-Life_of_Radioactive_Substance | https://proofwiki.org/wiki/Half-Life_of_Radioactive_Substance | [
"Half-Life",
"Radioactive Decay"
] | [
"Definition:Radioactive Decay/Radioactive Element",
"Definition:Radioactive Decay",
"Definition:First-Order Reaction/Rate Constant",
"Definition:Radioactive Decay/Half-Life",
"File:HalfLife.png"
] | [
"Definition:Time",
"Definition:Time",
"First-Order Reaction"
] |
proofwiki-10906 | Density not greater than Weight | Let $T = \struct {S, \tau}$ be a topological space.
Then
:$\map d T \le \map w T$
where
:$\map d T$ denotes the density of $T$,
:$\map w T$ denotes the weight of $T$. | By definition of weight there exists a basis $\BB$ of $T$:
:$\map w T = \card \BB$
where $\card \BB$ denotes the cardinality of $\BB$.
By Axiom of Choice define a mapping $f: \paren {\BB \setminus \O} \to S$:
:$\forall U \in \BB: U \ne \O \implies f \sqbrk U \in U$
We will prove that
:$\forall U \in \tau: U \ne \O \imp... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Then
:$\map d T \le \map w T$
where
:$\map d T$ denotes the [[Definition:Density of Topological Space|density]] of $T$,
:$\map w T$ denotes the [[Definition:Weight of Topological Space|weight]] of $T$. | By definition of [[Definition:Weight of Topological Space|weight]] there exists a [[Definition:Analytic Basis|basis]] $\BB$ of $T$:
:$\map w T = \card \BB$
where $\card \BB$ denotes the [[Definition:Cardinality|cardinality]] of $\BB$.
By [[Axiom:Axiom of Choice|Axiom of Choice]] define a [[Definition:Mapping|mapping]]... | Density not greater than Weight | https://proofwiki.org/wiki/Density_not_greater_than_Weight | https://proofwiki.org/wiki/Density_not_greater_than_Weight | [
"Denseness"
] | [
"Definition:Topological Space",
"Definition:Density of Topological Space",
"Definition:Weight of Topological Space"
] | [
"Definition:Weight of Topological Space",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Cardinality",
"Axiom:Axiom of Choice",
"Definition:Mapping",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Empty Set",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Image (S... |
proofwiki-10907 | Temperature of Body under Newton's Law of Cooling | Let $B$ be a body in an environment whose ambient temperature is $H_a$.
Let $H$ be the temperature of $B$ at time $t$.
Let $H_0$ be the temperature of $B$ at time $t = 0$.
Then:
:$H = H_a - \paren {H_0 - H_a} e^{-k t}$
where $k$ is some positive constant. | By Newton's Law of Cooling:
:The rate at which a hot body loses heat is proportional to the difference in temperature between it and its surroundings.
We have the differential equation:
:$\dfrac {\d H} {\d t} \propto - \paren {H - H_a}$
That is:
:$\dfrac {\d H} {\d t} = - k \paren {H - H_a}$
where $k$ is some positive ... | Let $B$ be a [[Definition:Body|body]] in an environment whose ambient [[Definition:Temperature|temperature]] is $H_a$.
Let $H$ be the [[Definition:Temperature|temperature]] of $B$ at time $t$.
Let $H_0$ be the [[Definition:Temperature|temperature]] of $B$ at time $t = 0$.
Then:
:$H = H_a - \paren {H_0 - H_a} e^{-k ... | By [[Newton's Law of Cooling]]:
:The [[Definition:Rate|rate]] at which a hot [[Definition:Body|body]] loses [[Definition:Heat|heat]] is [[Definition:Proportion|proportional]] to the difference in [[Definition:Temperature|temperature]] between it and its surroundings.
We have the [[Definition:First Order Ordinary Diff... | Temperature of Body under Newton's Law of Cooling | https://proofwiki.org/wiki/Temperature_of_Body_under_Newton's_Law_of_Cooling | https://proofwiki.org/wiki/Temperature_of_Body_under_Newton's_Law_of_Cooling | [
"Thermodynamics"
] | [
"Definition:Body",
"Definition:Temperature",
"Definition:Temperature",
"Definition:Temperature",
"Definition:Positive/Real Number",
"Definition:Constant"
] | [
"Newton's Law of Cooling",
"Definition:Rate",
"Definition:Body",
"Definition:Heat",
"Definition:Proportion",
"Definition:Temperature",
"Definition:First Order Ordinary Differential Equation",
"Definition:Positive/Real Number",
"Definition:Constant",
"Decay Equation"
] |
proofwiki-10908 | Speed of Body under Free Fall from Height | Let an object $B$ be released above ground from a point near the Earth's surface and allowed to fall freely.
Let $B$ fall a distance $s$.
Then:
:$v = \sqrt {2 g s}$
where:
:$v$ is the speed of $B$ after having fallen a distance $s$
:$g$ is the Acceleration Due to Gravity at the height through which $B$ falls.
It is sup... | From Equations of Motion with Constant Acceleration: Velocity after Distance:
:$\mathbf v \cdot \mathbf v = \mathbf u \cdot \mathbf u + 2 \mathbf g \cdot \mathbf s$
All dot products are between pairs of parallel vectors.
Thus by Cosine Formula for Dot Product:
:$v^2 = u^2 + 2 g s$
Here the body falls from rest, so:
:$\... | Let an [[Definition:Object|object]] $B$ be released above ground from a point near the [[Definition:Earth|Earth's]] surface and allowed to fall freely.
Let $B$ fall a [[Definition:Displacement|distance]] $s$.
Then:
:$v = \sqrt {2 g s}$
where:
:$v$ is the [[Definition:Speed|speed]] of $B$ after having fallen a [[Defin... | From [[Equations of Motion with Constant Acceleration/Velocity after Distance|Equations of Motion with Constant Acceleration: Velocity after Distance]]:
:$\mathbf v \cdot \mathbf v = \mathbf u \cdot \mathbf u + 2 \mathbf g \cdot \mathbf s$
All [[Definition:Dot Product|dot products]] are between pairs of [[Definition:... | Speed of Body under Free Fall from Height/Proof 1 | https://proofwiki.org/wiki/Speed_of_Body_under_Free_Fall_from_Height | https://proofwiki.org/wiki/Speed_of_Body_under_Free_Fall_from_Height/Proof_1 | [
"Mechanics",
"Gravity",
"Speed of Body under Free Fall from Height"
] | [
"Definition:Object",
"Definition:Earth",
"Definition:Displacement",
"Definition:Speed",
"Definition:Displacement",
"Acceleration Due to Gravity",
"Definition:Constant"
] | [
"Equations of Motion with Constant Acceleration/Velocity after Distance",
"Definition:Dot Product",
"Definition:Parallel (Geometry)/Lines",
"Definition:Vector",
"Cosine Formula for Dot Product",
"Definition:Stationary"
] |
proofwiki-10909 | Speed of Body under Free Fall from Height | Let an object $B$ be released above ground from a point near the Earth's surface and allowed to fall freely.
Let $B$ fall a distance $s$.
Then:
:$v = \sqrt {2 g s}$
where:
:$v$ is the speed of $B$ after having fallen a distance $s$
:$g$ is the Acceleration Due to Gravity at the height through which $B$ falls.
It is sup... | From Acceleration is Second Derivative of Displacement with respect to Time:
:$\mathbf g = \dfrac {\d^2 \mathbf s} {\d t^2}$
Integrating with respect to $t$, and by definition of velocity:
:$\mathbf v = \dfrac {\d \mathbf s} {\d t} = \mathbf g t + \mathbf c_1$
When $t = 0$, we have that $\mathrm c_1$ is the initial vel... | Let an [[Definition:Object|object]] $B$ be released above ground from a point near the [[Definition:Earth|Earth's]] surface and allowed to fall freely.
Let $B$ fall a [[Definition:Displacement|distance]] $s$.
Then:
:$v = \sqrt {2 g s}$
where:
:$v$ is the [[Definition:Speed|speed]] of $B$ after having fallen a [[Defin... | From [[Acceleration is Second Derivative of Displacement with respect to Time]]:
:$\mathbf g = \dfrac {\d^2 \mathbf s} {\d t^2}$
[[Definition:Integration|Integrating]] with respect to $t$, and by definition of [[Definition:Velocity|velocity]]:
:$\mathbf v = \dfrac {\d \mathbf s} {\d t} = \mathbf g t + \mathbf c_1$
W... | Speed of Body under Free Fall from Height/Proof 2 | https://proofwiki.org/wiki/Speed_of_Body_under_Free_Fall_from_Height | https://proofwiki.org/wiki/Speed_of_Body_under_Free_Fall_from_Height/Proof_2 | [
"Mechanics",
"Gravity",
"Speed of Body under Free Fall from Height"
] | [
"Definition:Object",
"Definition:Earth",
"Definition:Displacement",
"Definition:Speed",
"Definition:Displacement",
"Acceleration Due to Gravity",
"Definition:Constant"
] | [
"Acceleration is Second Derivative of Displacement with respect to Time",
"Definition:Primitive (Calculus)/Integration",
"Definition:Velocity",
"Definition:Velocity",
"Definition:Primitive (Calculus)/Integration",
"Definition:Displacement",
"Definition:Stationary"
] |
proofwiki-10910 | Speed of Body under Free Fall from Height | Let an object $B$ be released above ground from a point near the Earth's surface and allowed to fall freely.
Let $B$ fall a distance $s$.
Then:
:$v = \sqrt {2 g s}$
where:
:$v$ is the speed of $B$ after having fallen a distance $s$
:$g$ is the Acceleration Due to Gravity at the height through which $B$ falls.
It is sup... | From the Principle of Conservation of Energy:
:$K + P = C$
where:
:$K$ is the kinetic energy of $B$
:$P$ is the potential energy of $B$
:$C$ is a constant.
Let the mass of $B$ be $m$.
From Kinetic Energy of Motion:
:$K = \dfrac {m v^2} 2$
where $v$ is the speed of $B$.
From Potential Energy of Position:
:$P = m g s$
wh... | Let an [[Definition:Object|object]] $B$ be released above ground from a point near the [[Definition:Earth|Earth's]] surface and allowed to fall freely.
Let $B$ fall a [[Definition:Displacement|distance]] $s$.
Then:
:$v = \sqrt {2 g s}$
where:
:$v$ is the [[Definition:Speed|speed]] of $B$ after having fallen a [[Defin... | From the [[Principle of Conservation of Energy]]:
:$K + P = C$
where:
:$K$ is the [[Definition:Kinetic Energy|kinetic energy]] of $B$
:$P$ is the [[Definition:Potential Energy|potential energy]] of $B$
:$C$ is a [[Definition:Constant|constant]].
Let the [[Definition:Mass|mass]] of $B$ be $m$.
From [[Kinetic Energy of... | Speed of Body under Free Fall from Height/Proof 3 | https://proofwiki.org/wiki/Speed_of_Body_under_Free_Fall_from_Height | https://proofwiki.org/wiki/Speed_of_Body_under_Free_Fall_from_Height/Proof_3 | [
"Mechanics",
"Gravity",
"Speed of Body under Free Fall from Height"
] | [
"Definition:Object",
"Definition:Earth",
"Definition:Displacement",
"Definition:Speed",
"Definition:Displacement",
"Acceleration Due to Gravity",
"Definition:Constant"
] | [
"Principle of Conservation of Energy",
"Definition:Kinetic Energy",
"Definition:Potential Energy",
"Definition:Constant",
"Definition:Mass",
"Kinetic Energy of Motion",
"Definition:Speed",
"Potential Energy of Position",
"Definition:Distance",
"Definition:Stationary",
"Definition:Kinetic Energy"... |
proofwiki-10911 | Terminal Speed of Body under Fall Retarded Proportional to Speed | Let $B$ be a body falling in a gravitational field.
Let $B$ be falling through a medium which exerts a resisting force $k \mathbf v$ upon $B$ which is proportional to the speed of $B$ relative to the medium.
Then the terminal speed of $B$ is given by:
:$v = \dfrac {g m} k$ | Let $B$ start from rest.
From Motion of Body Falling through Air, the differential equation governing the motion of $B$ is given by:
:$m \dfrac {\d^2 \mathbf s} {\d t^2} = m \mathbf g - k \dfrac {\d \mathbf s} {\d t}$
Dividing through by $m$ and setting $c = \dfrac k m$ gives:
:$\dfrac {\d^2 \mathbf s} {\d t^2} = \math... | Let $B$ be a [[Definition:Body|body]] falling in a [[Definition:Gravitational Field|gravitational field]].
Let $B$ be falling through a medium which exerts a resisting [[Definition:Force|force]] $k \mathbf v$ upon $B$ which is [[Definition:Proportion|proportional]] to the [[Definition:Speed|speed]] of $B$ relative to ... | Let $B$ start from [[Definition:Stationary|rest]].
From [[Motion of Body Falling through Air]], the [[Definition:Differential Equation|differential equation]] governing the motion of $B$ is given by:
:$m \dfrac {\d^2 \mathbf s} {\d t^2} = m \mathbf g - k \dfrac {\d \mathbf s} {\d t}$
Dividing through by $m$ and sett... | Terminal Speed of Body under Fall Retarded Proportional to Speed | https://proofwiki.org/wiki/Terminal_Speed_of_Body_under_Fall_Retarded_Proportional_to_Speed | https://proofwiki.org/wiki/Terminal_Speed_of_Body_under_Fall_Retarded_Proportional_to_Speed | [
"Terminal Speed",
"Mechanics"
] | [
"Definition:Body",
"Definition:Gravitational Field",
"Definition:Force",
"Definition:Proportion",
"Definition:Speed",
"Definition:Terminal Speed"
] | [
"Definition:Stationary",
"Motion of Body Falling through Air",
"Definition:Differential Equation",
"Definition:Velocity",
"Definition:Magnitude"
] |
proofwiki-10912 | Approximate Motion of Simple Pendulum | Consider a simple pendulum consisting of a bob whose mass is $m$, at the end of a rod of negligible mass of length $a$.
Let the bob be pulled to one side so that the rod is at a small angle $\alpha$ (less than about $10 \degrees$ or $15 \degrees$) from the vertical and then released.
Let $T$ be the period of the pendul... | At a time $t$, let:
:the rod be at an angle $\theta$ to the the vertical
:the bob be travelling at a speed $v$
:the displacement of the bob from where it is when the rod is vertical, along its line of travel, be $s$.
:350px
From Motion of Simple Pendulum, the equation of motion of the bob is given by:
:$\dfrac {a^2} 2 ... | Consider a [[Definition:Simple Pendulum|simple pendulum]] consisting of a [[Definition:Pendulum Bob|bob]] whose [[Definition:Mass|mass]] is $m$, at the end of a [[Definition:Rod|rod]] of negligible [[Definition:Mass|mass]] of [[Definition:Length (Linear Measure)|length]] $a$.
Let the [[Definition:Pendulum Bob|bob]] be... | At a time $t$, let:
:the [[Definition:Rod|rod]] be at an [[Definition:Angle|angle]] $\theta$ to the [[Definition:Vertical|the vertical]]
:the [[Definition:Pendulum Bob|bob]] be travelling at a [[Definition:Speed|speed]] $v$
:the [[Definition:Displacement|displacement]] of the [[Definition:Pendulum Bob|bob]] from where ... | Approximate Motion of Simple Pendulum | https://proofwiki.org/wiki/Approximate_Motion_of_Simple_Pendulum | https://proofwiki.org/wiki/Approximate_Motion_of_Simple_Pendulum | [
"Simple Pendulums",
"Mechanics"
] | [
"Definition:Pendulum/Simple",
"Definition:Simple Pendulum/Bob",
"Definition:Mass",
"Definition:Rod",
"Definition:Mass",
"Definition:Linear Measure/Length",
"Definition:Simple Pendulum/Bob",
"Definition:Rod",
"Definition:Angle",
"Definition:Vertical",
"Definition:Simple Pendulum/Period",
"Defin... | [
"Definition:Rod",
"Definition:Angle",
"Definition:Vertical",
"Definition:Simple Pendulum/Bob",
"Definition:Speed",
"Definition:Displacement",
"Definition:Simple Pendulum/Bob",
"Definition:Rod",
"Definition:Vertical Line",
"File:MotionOfPendulum.png",
"Motion of Simple Pendulum",
"Definition:Si... |
proofwiki-10913 | Space is Separable iff Density not greater than Aleph Zero | Let $T$ be a topological space.
Then:
:$T$ is separable {{iff}} $\map d T \le \aleph_0$
where
:$\map d T$ denotes the density of $T$,
:$\aleph$ denotes the aleph mapping. | :$T$ is separable
{{iff}}:
:there exists a countable subset of $T$ which is dense by definition of separable space
{{iff}}:
:there exists a subset $A$ of $T$ such that $A$ is dense and exists an injection $A \to \N$ by definition of countable set
{{iff}}:
:there exists a subset $A$ of $T$ such that $A$ is dense and $\c... | Let $T$ be a [[Definition:Topological Space|topological space]].
Then:
:$T$ is [[Definition:Separable Space|separable]] {{iff}} $\map d T \le \aleph_0$
where
:$\map d T$ denotes the [[Definition:Density of Topological Space|density]] of $T$,
:$\aleph$ denotes the [[Definition:Aleph Mapping|aleph mapping]]. | :$T$ is [[Definition:Separable Space|separable]]
{{iff}}:
:there exists a [[Definition:Countable Set|countable]] [[Definition:Subset|subset]] of $T$ which is [[Definition:Everywhere Dense|dense]] by definition of [[Definition:Separable Space|separable space]]
{{iff}}:
:there exists a [[Definition:Subset|subset]] $A$ of... | Space is Separable iff Density not greater than Aleph Zero | https://proofwiki.org/wiki/Space_is_Separable_iff_Density_not_greater_than_Aleph_Zero | https://proofwiki.org/wiki/Space_is_Separable_iff_Density_not_greater_than_Aleph_Zero | [
"Denseness",
"Separable Spaces"
] | [
"Definition:Topological Space",
"Definition:Separable Space",
"Definition:Density of Topological Space",
"Definition:Aleph Mapping"
] | [
"Definition:Separable Space",
"Definition:Countable Set",
"Definition:Subset",
"Definition:Everywhere Dense",
"Definition:Separable Space",
"Definition:Subset",
"Definition:Everywhere Dense",
"Definition:Injection",
"Definition:Countable Set",
"Definition:Subset",
"Definition:Everywhere Dense",
... |
proofwiki-10914 | Boundary of Union of Separated Sets equals Union of Boundaries | Let $T$ be a topological space.
Let $A, B$ be subsets of $T$.
Let $A$ and $B$ are separated.
Then:
:$\map \partial {A \cup B} = \partial A \cup \partial B$
where:
:$\partial A$ denotes the boundary of $A$
:$A \cup B$ denotes the union of $A$ and $B$. | By definition of separated sets:
:$(1): \quad A^- \cap B = A \cap B^- = \O$
By Separated Sets are Disjoint:
:$A \cap B = \O$
{{begin-eqn}}
{{eqn | l = \partial A \cup \partial B
| r = \map \partial {A \cup B} \cup \map \partial {A \cap B} \cup \paren {\partial A \cap \partial B}
| c = Union of Boundaries
}}... | Let $T$ be a [[Definition:Topological Space|topological space]].
Let $A, B$ be [[Definition:Subset|subsets]] of $T$.
Let $A$ and $B$ are [[Definition:Separated Sets|separated]].
Then:
:$\map \partial {A \cup B} = \partial A \cup \partial B$
where:
:$\partial A$ denotes the [[Definition:Boundary (Topology)|boundary]]... | By definition of [[Definition:Separated Sets|separated sets]]:
:$(1): \quad A^- \cap B = A \cap B^- = \O$
By [[Separated Sets are Disjoint]]:
:$A \cap B = \O$
{{begin-eqn}}
{{eqn | l = \partial A \cup \partial B
| r = \map \partial {A \cup B} \cup \map \partial {A \cap B} \cup \paren {\partial A \cap \partial B... | Boundary of Union of Separated Sets equals Union of Boundaries | https://proofwiki.org/wiki/Boundary_of_Union_of_Separated_Sets_equals_Union_of_Boundaries | https://proofwiki.org/wiki/Boundary_of_Union_of_Separated_Sets_equals_Union_of_Boundaries | [
"Separated Sets",
"Set Boundaries"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Separated Sets",
"Definition:Boundary (Topology)",
"Definition:Set Union"
] | [
"Definition:Separated Sets",
"Separated Sets are Disjoint",
"Union of Boundaries",
"Boundary of Empty Set is Empty",
"Union with Empty Set",
"Definition:Set Intersection",
"Boundary is Intersection of Closure with Closure of Complement",
"Definition:Closure (Topology)",
"Definition:Relative Compleme... |
proofwiki-10915 | Boundary of Empty Set is Empty | Let $T$ be a topological space.
Then:
:$\partial_T \O = \O$
where $\partial_T \O$ denotes the boundary in topology $T$ of $\O$. | By Boundary is Intersection of Closure with Closure of Complement:
:$\partial_T \O = \O^- \cap \relcomp T \O^-$
where $\O^-$ denotes the closure of $\O$.
By Closure of Empty Set is Empty Set:
:$\O^- = \O$
Thus the result follows by Intersection with Empty Set.
{{qed}} | Let $T$ be a [[Definition:Topological Space|topological space]].
Then:
:$\partial_T \O = \O$
where $\partial_T \O$ denotes the [[Definition:Boundary (Topology)|boundary]] in topology $T$ of $\O$. | By [[Boundary is Intersection of Closure with Closure of Complement]]:
:$\partial_T \O = \O^- \cap \relcomp T \O^-$
where $\O^-$ denotes the [[Definition:Closure (Topology)|closure]] of $\O$.
By [[Closure of Empty Set is Empty Set]]:
:$\O^- = \O$
Thus the result follows by [[Intersection with Empty Set]].
{{qed}} | Boundary of Empty Set is Empty/Proof 1 | https://proofwiki.org/wiki/Boundary_of_Empty_Set_is_Empty | https://proofwiki.org/wiki/Boundary_of_Empty_Set_is_Empty/Proof_1 | [
"Set Boundaries",
"Boundary of Empty Set is Empty"
] | [
"Definition:Topological Space",
"Definition:Boundary (Topology)"
] | [
"Boundary is Intersection of Closure with Closure of Complement",
"Definition:Closure (Topology)",
"Closure of Empty Set is Empty Set",
"Intersection with Empty Set"
] |
proofwiki-10916 | Boundary of Empty Set is Empty | Let $T$ be a topological space.
Then:
:$\partial_T \O = \O$
where $\partial_T \O$ denotes the boundary in topology $T$ of $\O$. | From Open and Closed Sets in Topological Space, $\O$ is clopen in $T$.
The result follows from Set is Clopen iff Boundary is Empty.
{{qed}} | Let $T$ be a [[Definition:Topological Space|topological space]].
Then:
:$\partial_T \O = \O$
where $\partial_T \O$ denotes the [[Definition:Boundary (Topology)|boundary]] in topology $T$ of $\O$. | From [[Open and Closed Sets in Topological Space]], $\O$ is [[Definition:Clopen Set|clopen]] in $T$.
The result follows from [[Set is Clopen iff Boundary is Empty]].
{{qed}} | Boundary of Empty Set is Empty/Proof 2 | https://proofwiki.org/wiki/Boundary_of_Empty_Set_is_Empty | https://proofwiki.org/wiki/Boundary_of_Empty_Set_is_Empty/Proof_2 | [
"Set Boundaries",
"Boundary of Empty Set is Empty"
] | [
"Definition:Topological Space",
"Definition:Boundary (Topology)"
] | [
"Open and Closed Sets in Topological Space",
"Definition:Clopen Set",
"Set is Clopen iff Boundary is Empty"
] |
proofwiki-10917 | Terminal Speed of Body under Fall Retarded Proportional to Square of Speed | Let $B$ be a body falling in a gravitational field.
Let $B$ be falling through a medium which exerts a resisting force of magnitude $k v^2$ upon $B$ which is proportional to the square of the speed of $B$ relative to the medium.
Then the terminal speed of $B$ is given by:
:$v = \sqrt {\dfrac {g m} k}$ | Let $B$ start from rest.
The differential equation governing the motion of $B$ is given by:
:$m \dfrac {\d^2 \mathbf s} {\d t^2} = m \mathbf g - k \paren {\dfrac {\d \mathbf s} {\d t} }^2$
Dividing through by $m$ and setting $c = \dfrac k m$ gives:
:$\dfrac {\d^2 \mathbf s} {\d t^2} = \mathbf g - c \paren {\dfrac {\d \... | Let $B$ be a [[Definition:Body|body]] falling in a [[Definition:Gravitational Field|gravitational field]].
Let $B$ be falling through a medium which exerts a resisting [[Definition:Force|force]] of [[Definition:Magnitude|magnitude]] $k v^2$ upon $B$ which is [[Definition:Proportion|proportional]] to the [[Definition:S... | Let $B$ start from [[Definition:Stationary|rest]].
The [[Definition:Differential Equation|differential equation]] governing the motion of $B$ is given by:
:$m \dfrac {\d^2 \mathbf s} {\d t^2} = m \mathbf g - k \paren {\dfrac {\d \mathbf s} {\d t} }^2$
Dividing through by $m$ and setting $c = \dfrac k m$ gives:
:$\d... | Terminal Speed of Body under Fall Retarded Proportional to Square of Speed | https://proofwiki.org/wiki/Terminal_Speed_of_Body_under_Fall_Retarded_Proportional_to_Square_of_Speed | https://proofwiki.org/wiki/Terminal_Speed_of_Body_under_Fall_Retarded_Proportional_to_Square_of_Speed | [
"Terminal Speed",
"Mechanics"
] | [
"Definition:Body",
"Definition:Gravitational Field",
"Definition:Force",
"Definition:Magnitude",
"Definition:Proportion",
"Definition:Square/Function",
"Definition:Speed",
"Definition:Terminal Speed"
] | [
"Definition:Stationary",
"Definition:Differential Equation",
"Definition:Velocity",
"Definition:Constant",
"Definition:Magnitude",
"Primitive of Reciprocal of a squared minus x squared/Logarithm Form"
] |
proofwiki-10918 | Escape Speed of Projectile Fired Upwards | Let $P$ be a planet.
Let $P$ have an acceleration due to gravity at its surface of $g$.
Let $P$ have a radius of $R$.
Then the escape speed of $P$ is given by:
:$V = \sqrt {2 g R}$ | Let a projectile $B$ of mass $m$ be fired vertically upwards from the surface of $P$ at such a speed that it escapes the gravitational field of $P$ completely.
$F$ be the force exerted on $B$ by the gravitational field of $P$.
Let $x$ be the distance of $B$ from the surface of $P$ at time $t$.
We have:
:$F = -\dfrac k ... | Let $P$ be a [[Definition:Planet|planet]].
Let $P$ have an [[Acceleration Due to Gravity|acceleration due to gravity]] at its [[Definition:Surface|surface]] of $g$.
Let $P$ have a [[Definition:Radius of Sphere|radius]] of $R$.
Then the [[Definition:Escape Speed|escape speed]] of $P$ is given by:
:$V = \sqrt {2 g R}... | Let a [[Definition:Projectile|projectile]] $B$ of [[Definition:Mass|mass]] $m$ be fired [[Definition:Vertical Line|vertically]] upwards from the [[Definition:Surface|surface]] of $P$ at such a [[Definition:Speed|speed]] that it escapes the [[Definition:Gravitational Field|gravitational field]] of $P$ completely.
$F$ b... | Escape Speed of Projectile Fired Upwards/Proof 1 | https://proofwiki.org/wiki/Escape_Speed_of_Projectile_Fired_Upwards | https://proofwiki.org/wiki/Escape_Speed_of_Projectile_Fired_Upwards/Proof_1 | [
"Escape Speed of Projectile Fired Upwards",
"Escape Speed",
"Projectiles",
"Ballistics"
] | [
"Definition:Planet",
"Acceleration Due to Gravity",
"Definition:Surface",
"Definition:Sphere/Geometry/Radius",
"Definition:Escape Speed"
] | [
"Definition:Projectile",
"Definition:Mass",
"Definition:Vertical Line",
"Definition:Surface",
"Definition:Speed",
"Definition:Gravitational Field",
"Definition:Force",
"Definition:Gravitational Field",
"Definition:Time",
"Definition:Speed",
"Definition:Time",
"Definition:Speed"
] |
proofwiki-10919 | Union of Boundaries | Let $T = \struct {S, \tau}$ be a topological space.
Let $A, B$ be subsets of $S$.
Then:
:$\partial A \cup \partial B = \map \partial {A \cup B} \cup \map \partial {A \cap B} \cup \paren {\partial A \cap \partial B}$
where $\partial A$ denotes the boundary of $A$. | First we will prove that
:$\partial A \subseteq \map \partial {A \cup B} \cup \map \partial {A \cap B} \cup \paren {\partial A \cap \partial B}$
Let $x \in \partial A$.
{{AimForCont}} that
:$x \notin \map \partial {A \cup B} \cup \map \partial {A \cap B} \cup \paren {\partial A \cap \partial B}$
Then by definition of u... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A, B$ be [[Definition:Subset|subsets]] of $S$.
Then:
:$\partial A \cup \partial B = \map \partial {A \cup B} \cup \map \partial {A \cap B} \cup \paren {\partial A \cap \partial B}$
where $\partial A$ denotes the [[Definition:Bou... | First we will prove that
:$\partial A \subseteq \map \partial {A \cup B} \cup \map \partial {A \cap B} \cup \paren {\partial A \cap \partial B}$
Let $x \in \partial A$.
{{AimForCont}} that
:$x \notin \map \partial {A \cup B} \cup \map \partial {A \cap B} \cup \paren {\partial A \cap \partial B}$
Then by definition o... | Union of Boundaries | https://proofwiki.org/wiki/Union_of_Boundaries | https://proofwiki.org/wiki/Union_of_Boundaries | [
"Set Boundaries"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Boundary (Topology)"
] | [
"Definition:Set Union",
"Characterization of Boundary by Open Sets",
"Intersection Distributes over Union",
"De Morgan's Laws (Set Theory)/Relative Complement/Complement of Union",
"Definition:Set Intersection",
"Characterization of Boundary by Open Sets",
"Characterization of Boundary by Open Sets",
... |
proofwiki-10920 | Length of Arch of Sine Function | The length of one arch of the sine function:
:$y = \sin x$
is given by:
:$L = 2 \sqrt 2 \map E {\dfrac {\sqrt 2} 2}$
where $E$ denotes the incomplete elliptic integral of the second kind. | Let $L$ be the length of one arch of $y = \sin x$.
Then:
{{begin-eqn}}
{{eqn | l = L
| r = 2 \int_0^{\pi/2} \sqrt {1 + \paren {\map {\frac \d {\d x} } {\sin x} }^2} \rd x
| c = {{Defof|Length of Curve}}
}}
{{eqn | r = 2 \int_0^{\pi/2} \sqrt {1 + \cos^2 x} \rd x
| c = Derivative of Sine Function
}}
{{e... | The [[Definition:Length of Curve|length]] of one [[Definition:Arch of Sine Function|arch]] of the [[Definition:Sine Function|sine function]]:
:$y = \sin x$
is given by:
:$L = 2 \sqrt 2 \map E {\dfrac {\sqrt 2} 2}$
where $E$ denotes the [[Definition:Complete Elliptic Integral of the Second Kind|incomplete elliptic int... | Let $L$ be the [[Definition:Length of Curve|length]] of one [[Definition:Arch of Sine Function|arch]] of $y = \sin x$.
Then:
{{begin-eqn}}
{{eqn | l = L
| r = 2 \int_0^{\pi/2} \sqrt {1 + \paren {\map {\frac \d {\d x} } {\sin x} }^2} \rd x
| c = {{Defof|Length of Curve}}
}}
{{eqn | r = 2 \int_0^{\pi/2} \sq... | Length of Arch of Sine Function | https://proofwiki.org/wiki/Length_of_Arch_of_Sine_Function | https://proofwiki.org/wiki/Length_of_Arch_of_Sine_Function | [
"Sine Function",
"Complete Elliptic Integral of the Second Kind"
] | [
"Definition:Arc Length",
"Definition:Sine/Real Function/Arch",
"Definition:Sine",
"Definition:Elliptic Integral of the Second Kind/Complete"
] | [
"Definition:Arc Length",
"Definition:Sine/Real Function/Arch",
"Derivative of Sine Function",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-10921 | Length of Lemniscate of Bernoulli | The total length of the lemniscate of Bernoulli given in polar coordinates as:
:$r^2 = a^2 \cos 2 \theta$
is given by:
{{begin-eqn}}
{{eqn | l = L
| r = 4 a \map F {\sqrt 2, \dfrac \pi 4}
| c =
}}
{{eqn | r = \dfrac 1 {\sqrt {2 \pi} } \paren {\map \Gamma {\dfrac 1 4} }^2
| c =
}}
{{end-eqn}}
where $... | The arc length of a small length increment $\d s$ is given in polar co-ordinates by:
:$\paren {\d s}^2 = \paren {r \rd \theta}^2 + \paren {\d r}^2$
from which:
:$\dfrac {\d s} {\d \theta} = \sqrt {r^2 + \paren {\dfrac {\d r} {\d \theta} }^2}$
Half of one lobe of the lemniscate is achieved when $\theta$ goes from $0$ to... | The total [[Definition:Length of Curve|length]] of the [[Definition:Lemniscate of Bernoulli|lemniscate of Bernoulli]] given in [[Definition:Polar Coordinates|polar coordinates]] as:
:$r^2 = a^2 \cos 2 \theta$
is given by:
{{begin-eqn}}
{{eqn | l = L
| r = 4 a \map F {\sqrt 2, \dfrac \pi 4}
| c =
}}
{{eqn ... | The [[Definition:Length of Curve|arc length]] of a small length increment $\d s$ is given in [[Definition:Polar Coordinates|polar co-ordinates]] by:
:$\paren {\d s}^2 = \paren {r \rd \theta}^2 + \paren {\d r}^2$
from which:
:$\dfrac {\d s} {\d \theta} = \sqrt {r^2 + \paren {\dfrac {\d r} {\d \theta} }^2}$
Half of on... | Length of Lemniscate of Bernoulli | https://proofwiki.org/wiki/Length_of_Lemniscate_of_Bernoulli | https://proofwiki.org/wiki/Length_of_Lemniscate_of_Bernoulli | [
"Lemniscate of Bernoulli",
"Incomplete Elliptic Integral of the First Kind"
] | [
"Definition:Arc Length",
"Definition:Lemniscate of Bernoulli",
"Definition:Polar Coordinates",
"Definition:Elliptic Integral of the First Kind/Incomplete"
] | [
"Definition:Arc Length",
"Definition:Polar Coordinates",
"Definition:Lemniscate of Bernoulli/Lobe",
"Definition:Lemniscate of Bernoulli",
"Definition:Arc Length",
"Definition:Lemniscate of Bernoulli",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-10922 | Boundary of Union is Subset of Union of Boundaries | Let $T = \struct {S, \tau}$ be a topological space.
Let $A, B$ be subsets of $S$.
Then:
:$\map \partial {A \cup B} \subseteq \partial A \cup \partial B$
where $\partial A$ denotes the boundary of $A$. | By Intersection is Subset:
:$\relcomp S A \cap \relcomp S B \subseteq \relcomp S A \land \relcomp S A \cap \relcomp S B \subseteq \relcomp S B$
Then by Topological Closure of Subset is Subset of Topological Closure:
:$\paren {\relcomp S A \cap \relcomp S B}^- \subseteq \paren {\relcomp S A}^- \land \paren {\relcomp S A... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A, B$ be [[Definition:Subset|subsets]] of $S$.
Then:
:$\map \partial {A \cup B} \subseteq \partial A \cup \partial B$
where $\partial A$ denotes the [[Definition:Boundary (Topology)|boundary]] of $A$. | By [[Intersection is Subset]]:
:$\relcomp S A \cap \relcomp S B \subseteq \relcomp S A \land \relcomp S A \cap \relcomp S B \subseteq \relcomp S B$
Then by [[Topological Closure of Subset is Subset of Topological Closure]]:
:$\paren {\relcomp S A \cap \relcomp S B}^- \subseteq \paren {\relcomp S A}^- \land \paren {\re... | Boundary of Union is Subset of Union of Boundaries | https://proofwiki.org/wiki/Boundary_of_Union_is_Subset_of_Union_of_Boundaries | https://proofwiki.org/wiki/Boundary_of_Union_is_Subset_of_Union_of_Boundaries | [
"Set Boundaries"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Boundary (Topology)"
] | [
"Intersection is Subset",
"Topological Closure of Subset is Subset of Topological Closure",
"Boundary is Intersection of Closure with Closure of Complement",
"Boundary is Intersection of Closure with Closure of Complement",
"De Morgan's Laws (Set Theory)/Relative Complement/Complement of Union",
"Closure ... |
proofwiki-10923 | Boundary of Intersection is Subset of Union of Boundaries | Let $T = \struct {S, \tau}$ be a topological space.
Let $A, B$ be subsets of $S$.
Then:
:$\map \partial {A \cap B} \subseteq \partial A \cup \partial B$
where $\partial A$ denotes the boundary of $A$. | By Intersection is Subset:
:$A \cap B \subseteq A \land A \cap B \subseteq B$
Then by Topological Closure of Subset is Subset of Topological Closure:
:$\paren {A \cap B}^- \subseteq A^- \land \paren {A \cap B}^- \subseteq B^-$
Hence by Boundary is Intersection of Closure with Closure of Complement:
:$\paren {A \cap B}^... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A, B$ be [[Definition:Subset|subsets]] of $S$.
Then:
:$\map \partial {A \cap B} \subseteq \partial A \cup \partial B$
where $\partial A$ denotes the [[Definition:Boundary (Topology)|boundary]] of $A$. | By [[Intersection is Subset]]:
:$A \cap B \subseteq A \land A \cap B \subseteq B$
Then by [[Topological Closure of Subset is Subset of Topological Closure]]:
:$\paren {A \cap B}^- \subseteq A^- \land \paren {A \cap B}^- \subseteq B^-$
Hence by [[Boundary is Intersection of Closure with Closure of Complement]]:
:$\par... | Boundary of Intersection is Subset of Union of Boundaries | https://proofwiki.org/wiki/Boundary_of_Intersection_is_Subset_of_Union_of_Boundaries | https://proofwiki.org/wiki/Boundary_of_Intersection_is_Subset_of_Union_of_Boundaries | [
"Set Boundaries"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Boundary (Topology)"
] | [
"Intersection is Subset",
"Topological Closure of Subset is Subset of Topological Closure",
"Boundary is Intersection of Closure with Closure of Complement",
"Boundary is Intersection of Closure with Closure of Complement",
"De Morgan's Laws (Set Theory)/Relative Complement/Complement of Intersection",
"C... |
proofwiki-10924 | Complete Elliptic Integral of the First Kind as Power Series | The '''complete elliptic integral of the first kind''':
:$\ds \map K k = \int_0^{\pi / 2} \frac {\rd \phi} {\sqrt {1 - k^2 \sin^2 \phi} } = \int_0^1 \frac {\rd v} {\sqrt {\paren {1 - v^2} \paren {1 - k^2 v^2} } }$
can be expressed as the power series:
{{begin-eqn}}
{{eqn | l = \map K k
| r = \frac \pi 2 \sum_{i \... | From Reduction Formula for Integral of Power of Sine, $\forall i \in \N$:
{{begin-eqn}}
{{eqn | l = \int_0^{\pi / 2} \sin^{2 i} \phi \rd \phi
| r = \frac {2 i - 1} {2 i} \int_0^{\pi / 2} \sin^{2 i - 2} \phi \rd \phi - \intlimits {\frac {\sin^{2 i - 1} x \cos x} i} {x = 0} {x = \frac \pi 2}
}}
{{eqn | r = \frac {2... | The '''[[Definition:Complete Elliptic Integral of the First Kind|complete elliptic integral of the first kind]]''':
:$\ds \map K k = \int_0^{\pi / 2} \frac {\rd \phi} {\sqrt {1 - k^2 \sin^2 \phi} } = \int_0^1 \frac {\rd v} {\sqrt {\paren {1 - v^2} \paren {1 - k^2 v^2} } }$
can be expressed as the [[Definition:Power Se... | From [[Reduction Formula for Integral of Power of Sine]], $\forall i \in \N$:
{{begin-eqn}}
{{eqn | l = \int_0^{\pi / 2} \sin^{2 i} \phi \rd \phi
| r = \frac {2 i - 1} {2 i} \int_0^{\pi / 2} \sin^{2 i - 2} \phi \rd \phi - \intlimits {\frac {\sin^{2 i - 1} x \cos x} i} {x = 0} {x = \frac \pi 2}
}}
{{eqn | r = \fr... | Complete Elliptic Integral of the First Kind as Power Series | https://proofwiki.org/wiki/Complete_Elliptic_Integral_of_the_First_Kind_as_Power_Series | https://proofwiki.org/wiki/Complete_Elliptic_Integral_of_the_First_Kind_as_Power_Series | [
"Complete Elliptic Integral of the First Kind"
] | [
"Definition:Elliptic Integral of the First Kind/Complete",
"Definition:Power Series"
] | [
"Reduction Formula for Integral of Power of Sine",
"Binomial Theorem/General Binomial Theorem"
] |
proofwiki-10925 | Complete Elliptic Integral of the Second Kind as Power Series | The '''complete elliptic integral of the second kind''':
:$\ds \map E k = \int_0^{\pi / 2} \sqrt {1 - k^2 \sin^2 \phi} \, \rd \phi = \int_0^1 \dfrac {\sqrt {1 - k^2 v^2} } {\sqrt {1 - v^2}} \, \rd v$
can be expressed as the power series:
{{begin-eqn}}
{{eqn | l = \map E k
| r = \frac \pi 2 \sum_{i \mathop \ge 0} ... | From Reduction Formula for Integral of Power of Sine, $\forall i \in \N$:
{{begin-eqn}}
{{eqn | l = \int_0^{\pi / 2} \sin^{2 i} \phi \rd \phi
| r = \frac {2 i - 1} {2 i} \int_0^{\pi / 2} \sin^{2 i - 2} \phi \rd \phi - \intlimits {\frac {\sin^{2 i - 1} x \cos x} i} {x = 0} {x = \frac \pi 2}
}}
{{eqn | r = \frac {2... | The '''[[Definition:Complete Elliptic Integral of the Second Kind|complete elliptic integral of the second kind]]''':
:$\ds \map E k = \int_0^{\pi / 2} \sqrt {1 - k^2 \sin^2 \phi} \, \rd \phi = \int_0^1 \dfrac {\sqrt {1 - k^2 v^2} } {\sqrt {1 - v^2}} \, \rd v$
can be expressed as the [[Definition:Power Series|power se... | From [[Reduction Formula for Integral of Power of Sine]], $\forall i \in \N$:
{{begin-eqn}}
{{eqn | l = \int_0^{\pi / 2} \sin^{2 i} \phi \rd \phi
| r = \frac {2 i - 1} {2 i} \int_0^{\pi / 2} \sin^{2 i - 2} \phi \rd \phi - \intlimits {\frac {\sin^{2 i - 1} x \cos x} i} {x = 0} {x = \frac \pi 2}
}}
{{eqn | r = \fr... | Complete Elliptic Integral of the Second Kind as Power Series | https://proofwiki.org/wiki/Complete_Elliptic_Integral_of_the_Second_Kind_as_Power_Series | https://proofwiki.org/wiki/Complete_Elliptic_Integral_of_the_Second_Kind_as_Power_Series | [
"Complete Elliptic Integral of the Second Kind"
] | [
"Definition:Elliptic Integral of the Second Kind/Complete",
"Definition:Power Series"
] | [
"Reduction Formula for Integral of Power of Sine",
"Binomial Theorem/General Binomial Theorem"
] |
proofwiki-10926 | Discrete Space is Separable iff Countable | Let $T = \struct {S, \tau}$ be a discrete topological space.
Then:
:$T$ is separable {{iff}} $S$ is countable. | === Sufficient Condition ===
Immediate from Separable Discrete Space is Countable.
{{qed|lemma}} | Let $T = \struct {S, \tau}$ be a [[Definition:Discrete Space|discrete]] [[Definition:Topological Space|topological space]].
Then:
:$T$ is [[Definition:Separable Space|separable]] {{iff}} $S$ is [[Definition:Countable Set|countable]]. | === Sufficient Condition ===
Immediate from [[Separable Discrete Space is Countable]].
{{qed|lemma}} | Discrete Space is Separable iff Countable | https://proofwiki.org/wiki/Discrete_Space_is_Separable_iff_Countable | https://proofwiki.org/wiki/Discrete_Space_is_Separable_iff_Countable | [
"Discrete Topologies",
"Examples of Separable Spaces"
] | [
"Definition:Discrete Topology",
"Definition:Topological Space",
"Definition:Separable Space",
"Definition:Countable Set"
] | [
"Separable Discrete Space is Countable"
] |
proofwiki-10927 | Set is Countable iff Cardinality not greater Aleph Zero | Let $X$ be a set.
Then:
:$X$ is countable {{iff}} $\size X \le \aleph_0$
where
:$\size X$ denotes the cardinality of $X$,
:$\aleph$ denotes the aleph mapping. | :$X$ is countable
{{iff}}
:there exists an injection $X \to \N$ by definition of countable set
{{iff}}
:$\size X \le \size \N$ by Injection iff Cardinal Inequality
{{iff}}
:$\size X \le \aleph_0$ by Aleph Zero equals Cardinality of Naturals.
{{qed}}
Category:Countable Sets
odwikgixvfndjekh3c1vm2d04dez9q6 | Let $X$ be a [[Definition:Set|set]].
Then:
:$X$ is [[Definition:Countable Set|countable]] {{iff}} $\size X \le \aleph_0$
where
:$\size X$ denotes the [[Definition:Cardinality|cardinality]] of $X$,
:$\aleph$ denotes the [[Definition:Aleph Mapping|aleph mapping]]. | :$X$ is [[Definition:Countable Set|countable]]
{{iff}}
:there exists an [[Definition:Injection|injection]] $X \to \N$ by definition of [[Definition:Countable Set|countable set]]
{{iff}}
:$\size X \le \size \N$ by [[Injection iff Cardinal Inequality]]
{{iff}}
:$\size X \le \aleph_0$ by [[Aleph Zero equals Cardinality of... | Set is Countable iff Cardinality not greater Aleph Zero | https://proofwiki.org/wiki/Set_is_Countable_iff_Cardinality_not_greater_Aleph_Zero | https://proofwiki.org/wiki/Set_is_Countable_iff_Cardinality_not_greater_Aleph_Zero | [
"Countable Sets"
] | [
"Definition:Set",
"Definition:Countable Set",
"Definition:Cardinality",
"Definition:Aleph Mapping"
] | [
"Definition:Countable Set",
"Definition:Injection",
"Definition:Countable Set",
"Injection iff Cardinal Inequality",
"Aleph Zero equals Cardinality of Naturals",
"Category:Countable Sets"
] |
proofwiki-10928 | Bernoulli's Theorem | Let the probability of the occurrence of an event be $p$.
Let $n$ independent trials be made, with $k$ successes.
Then:
:$\ds \lim_{n \mathop \to \infty} \frac k n = p$ | Let the random variable $k$ have the binomial distribution with parameters $n$ and $p$, that is:
:$k \sim \Binomial n p$
where $k$ denotes the number of successes of the $n$ independent trials of the event with probability $p$.
From Expectation of Binomial Distribution:
:$\expect k = n p \leadsto \dfrac 1 n \expect k =... | Let the [[Definition:Probability|probability]] of the [[Definition:Occurrence of Event|occurrence]] of an [[Definition:Event|event]] be $p$.
Let $n$ [[Definition:Independent Events|independent trials]] be made, with $k$ [[Definition:Success|successes]].
Then:
:$\ds \lim_{n \mathop \to \infty} \frac k n = p$ | Let the [[Definition:Random Variable|random variable]] $k$ have the [[Definition:Binomial Distribution|binomial distribution]] with parameters $n$ and $p$, that is:
:$k \sim \Binomial n p$
where $k$ denotes the number of [[Definition:Success|successes]] of the $n$ [[Definition:Independent Events|independent trials]] of... | Bernoulli's Theorem | https://proofwiki.org/wiki/Bernoulli's_Theorem | https://proofwiki.org/wiki/Bernoulli's_Theorem | [
"Bernoulli's Theorem",
"Laws of Large Numbers",
"Probability Theory"
] | [
"Definition:Probability",
"Definition:Event/Occurrence",
"Definition:Event",
"Definition:Independent Events",
"Definition:Bernoulli Distribution"
] | [
"Definition:Random Variable",
"Definition:Binomial Distribution",
"Definition:Bernoulli Distribution",
"Definition:Independent Events",
"Definition:Probability",
"Expectation of Binomial Distribution",
"Expectation is Linear",
"Variance of Binomial Distribution",
"Variance of Linear Combination of R... |
proofwiki-10929 | Time of Travel down Brachistochrone | Let a wire $AB$ be curved into the shape of a brachistochrone.
Let $AB$ be embedded in a constant and uniform gravitational field where Acceleration Due to Gravity is $g$.
Let a bead $P$ be released at $A$ to slide down without friction to $B$.
Then the time taken for $P$ to slide from $A$ to $B$ is:
:$T = \pi \sqrt {\... | That the curve $AB$ is indeed a cycloid is demonstrated in Brachistochrone is Cycloid.
Let $A$ be located at the origin of a cartesian plane.
We have the equations of the cycloid:
{{begin-eqn}}
{{eqn | l = x
| r = a \paren {\theta - \sin \theta}
}}
{{eqn | l = y
| r = a \paren {1 - \cos \theta}
}}
{{end-eqn... | Let a [[Definition:Wire|wire]] $AB$ be curved into the shape of a [[Definition:Brachistochrone|brachistochrone]].
Let $AB$ be embedded in a [[Definition:Constant|constant]] and [[Definition:Uniform|uniform]] [[Definition:Gravitational Field|gravitational field]] where [[Acceleration Due to Gravity]] is $g$.
Let a [[D... | That the [[Definition:Curve|curve]] $AB$ is indeed a [[Definition:Cycloid|cycloid]] is demonstrated in [[Brachistochrone is Cycloid]].
Let $A$ be located at the [[Definition:Origin|origin]] of a [[Definition:Cartesian Plane|cartesian plane]].
We have the equations of the [[Definition:Cycloid|cycloid]]:
{{begin-eqn}... | Time of Travel down Brachistochrone | https://proofwiki.org/wiki/Time_of_Travel_down_Brachistochrone | https://proofwiki.org/wiki/Time_of_Travel_down_Brachistochrone | [
"Cycloids"
] | [
"Definition:Wire",
"Definition:Brachistochrone",
"Definition:Constant",
"Definition:Uniform",
"Definition:Gravitational Field",
"Acceleration Due to Gravity",
"Definition:Bead",
"Definition:Friction",
"Definition:Time",
"Definition:Circle/Radius",
"Definition:Cycloid/Generating Circle",
"Defin... | [
"Definition:Line/Curve",
"Definition:Cycloid",
"Brachistochrone is Cycloid",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Definition:Cycloid",
"Definition:Derivative/Real Function/With Respect To",
"Definition:Arc Distance",
"Definition:Coordinate System/Origin",
"Principl... |
proofwiki-10930 | Time of Travel down Brachistochrone/Corollary | Let a bead $P$ be released from anywhere on the wire between $A$ and $B$ to slide down without friction to $B$.
Then the time taken for $P$ to slide to $B$ is:
:$T = \pi \sqrt{\dfrac a g}$ | That the curve $AB$ is indeed a cycloid is demonstrated in Brachistochrone is Cycloid.
Let $A$ be located at the origin of a cartesian plane.
Let the bead slide from an intermediate point $\theta_0$.
We have:
:$v = \dfrac {\d s} {\d t} = \sqrt {2 g \paren {y - y_0} }$
which leads us, via the same route as for Time of T... | Let a [[Definition:Bead|bead]] $P$ be released from anywhere on the [[Definition:Wire|wire]] between $A$ and $B$ to slide down without [[Definition:Friction|friction]] to $B$.
Then the [[Definition:Time|time]] taken for $P$ to slide to $B$ is:
:$T = \pi \sqrt{\dfrac a g}$ | That the [[Definition:Curve|curve]] $AB$ is indeed a [[Definition:Cycloid|cycloid]] is demonstrated in [[Brachistochrone is Cycloid]].
Let $A$ be located at the [[Definition:Origin|origin]] of a [[Definition:Cartesian Plane|cartesian plane]].
Let the [[Definition:Bead|bead]] slide from an intermediate point $\theta_0... | Time of Travel down Brachistochrone/Corollary | https://proofwiki.org/wiki/Time_of_Travel_down_Brachistochrone/Corollary | https://proofwiki.org/wiki/Time_of_Travel_down_Brachistochrone/Corollary | [
"Cycloids"
] | [
"Definition:Bead",
"Definition:Wire",
"Definition:Friction",
"Definition:Time"
] | [
"Definition:Line/Curve",
"Definition:Cycloid",
"Brachistochrone is Cycloid",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Definition:Bead",
"Time of Travel down Brachistochrone",
"Half Angle Formulas/Cosine",
"Half Angle Formulas/Sine",
"Definition:Time",
"Definition:Bra... |
proofwiki-10931 | Rational Numbers are F-Sigma Set in Real Line | Let $\struct {\R, \tau}$ be the real number line considered asa topological space with the usual (Euclidean) topology.
Then:
:$\Q$ is an $F_\sigma$ set in $\struct {\R, \tau}$. | Define the set of subsets of $\R$ as:
:$\FF := \set {\set x: x \in \Q}$
By Closed Real Interval is Closed Set:
:$\forall x \in \Q: \closedint x x = \set x$ is closed (in topological sense)
Then:
:$\forall A \in \FF: A$ is closed
By Cardinality of Set of Singletons:
:$\card \FF = \card \Q$
where $\card \FF$ denotes the ... | Let $\struct {\R, \tau}$ be the [[Definition:Real Number Line|real number line]] considered asa [[Definition:Topological Space|topological space]] with the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]].
Then:
:$\Q$ is an [[Definition:F-Sigma Set|$F_\sigma$ set]] in $\struct {\R, \ta... | Define the [[Definition:Set of Sets|set]] of [[Definition:Subset|subsets]] of $\R$ as:
:$\FF := \set {\set x: x \in \Q}$
By [[Closed Real Interval is Closed Set]]:
:$\forall x \in \Q: \closedint x x = \set x$ is [[Definition:Closed Set (Topology)|closed (in topological sense)]]
Then:
:$\forall A \in \FF: A$ is [[Defi... | Rational Numbers are F-Sigma Set in Real Line | https://proofwiki.org/wiki/Rational_Numbers_are_F-Sigma_Set_in_Real_Line | https://proofwiki.org/wiki/Rational_Numbers_are_F-Sigma_Set_in_Real_Line | [
"F-Sigma Sets"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Topological Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:F-Sigma Set"
] | [
"Definition:Set of Sets",
"Definition:Subset",
"Closed Real Interval is Closed Set",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Cardinality of Set of Singletons",
"Definition:Cardinality",
"Rational Numbers are Countably Infinite",
"Definition:Countable Set",
"Set is Coun... |
proofwiki-10932 | Union of Set of Singletons | Let $S$ be a set.
Let $T = \set {\set x: x \in S}$ be the set of all singletons of elements of $S$.
Then:
:$\ds \bigcup T = S$
where $\ds \bigcup T$ denotes the union of $T$. | === Union of $T$ Subset $S$ ===
Let $\ds x \in \bigcup T$.
By definition of union:
:$\exists A \in T: x \in A$
By definition of $T$:
:$\exists y \in S: A = \set y$
Then by definition of singleton:
:$x = y$
Thus $x \in S$.
{{qed|lemma}} | Let $S$ be a [[Definition:Set|set]].
Let $T = \set {\set x: x \in S}$ be the [[Definition:Set|set]] of all [[Definition:Singleton|singletons]] of [[Definition:Element|elements]] of $S$.
Then:
:$\ds \bigcup T = S$
where $\ds \bigcup T$ denotes the [[Definition:Union of Set of Sets|union]] of $T$. | === Union of $T$ Subset $S$ ===
Let $\ds x \in \bigcup T$.
By definition of [[Definition:Union of Set of Sets|union]]:
:$\exists A \in T: x \in A$
By definition of $T$:
:$\exists y \in S: A = \set y$
Then by definition of [[Definition:Singleton|singleton]]:
:$x = y$
Thus $x \in S$.
{{qed|lemma}} | Union of Set of Singletons | https://proofwiki.org/wiki/Union_of_Set_of_Singletons | https://proofwiki.org/wiki/Union_of_Set_of_Singletons | [
"Set Union",
"Singletons"
] | [
"Definition:Set",
"Definition:Set",
"Definition:Singleton",
"Definition:Element",
"Definition:Set Union/Set of Sets"
] | [
"Definition:Set Union/Set of Sets",
"Definition:Singleton",
"Definition:Singleton"
] |
proofwiki-10933 | Confocal Conics are Self-Orthogonal | The confocal conics defined by:
:$\quad \dfrac {x^2} {a^2} + \dfrac {y^2} {a^2 - c^2} = 1$
forms a family of orthogonal trajectories which is self-orthogonal.
:500px | Consider:
:$(1): \quad \dfrac {x^2} {a^2} + \dfrac {y^2} {a^2 - c^2} = 1$
From Equation of Confocal Ellipses: Formulation 2:
:$(1)$ defines an ellipse when $a^2 > c^2$.
From Equation of Confocal Hyperbolas: Formulation 2:
:$(1)$ defines a hyperbola when $a^2 < c^2$.
Thus it is seen that $(1)$ is that of a conic section... | The [[Definition:Confocal Conics|confocal conics]] defined by:
:$\quad \dfrac {x^2} {a^2} + \dfrac {y^2} {a^2 - c^2} = 1$
forms a [[Definition:Orthogonal Trajectories|family of orthogonal trajectories]] which is [[Definition:Self-Orthogonal Trajectories|self-orthogonal]].
:[[File:ConfocalConics.png|500px]] | Consider:
:$(1): \quad \dfrac {x^2} {a^2} + \dfrac {y^2} {a^2 - c^2} = 1$
From [[Equation of Confocal Ellipses/Formulation 2|Equation of Confocal Ellipses: Formulation 2]]:
:$(1)$ defines an [[Definition:Ellipse|ellipse]] when $a^2 > c^2$.
From [[Equation of Confocal Hyperbolas/Formulation 2|Equation of Confocal Hype... | Confocal Conics are Self-Orthogonal | https://proofwiki.org/wiki/Confocal_Conics_are_Self-Orthogonal | https://proofwiki.org/wiki/Confocal_Conics_are_Self-Orthogonal | [
"Orthogonal Trajectories",
"Conic Sections",
"Orthogonal Curves"
] | [
"Definition:Confocal Conics",
"Definition:Orthogonal Trajectories",
"Definition:Self-Orthogonal Trajectories",
"File:ConfocalConics.png"
] | [
"Equation of Confocal Ellipses/Formulation 2",
"Definition:Ellipse",
"Equation of Confocal Hyperbolas/Formulation 2",
"Definition:Hyperbola",
"Definition:Conic Section",
"Definition:Formation of Ordinary Differential Equation by Elimination",
"Definition:Differentiation",
"Definition:Self-Orthogonal T... |
proofwiki-10934 | T1 Space is T1/2 Space | Let $T$ be a $T_1$ topological space.
Then $T$ is $T_{\frac 1 2}$ space. | By Closure of Derivative is Derivative in T1 Space:
:$\forall A \subseteq T: \paren {A'}^- = A'$
where
:$A'$ denotes the derivative of $A$
:$\paren {A'}^-$ denotes the closure of $A'$
Then by Topological Closure is Closed:
:$\forall A \subseteq T: A'$ is closed
Thus by definition:
:$T$ is $T_{\frac 1 2}$ space
{{qed}} | Let $T$ be a [[Definition:T1 Space|$T_1$]] [[Definition:Topological Space|topological space]].
Then $T$ is [[Definition:T1/2 Space|$T_{\frac 1 2}$ space]]. | By [[Closure of Derivative is Derivative in T1 Space]]:
:$\forall A \subseteq T: \paren {A'}^- = A'$
where
:$A'$ denotes the [[Definition:Set Derivative|derivative]] of $A$
:$\paren {A'}^-$ denotes the [[Definition:Closure (Topology)|closure]] of $A'$
Then by [[Topological Closure is Closed]]:
:$\forall A \subseteq T:... | T1 Space is T1/2 Space | https://proofwiki.org/wiki/T1_Space_is_T1/2_Space | https://proofwiki.org/wiki/T1_Space_is_T1/2_Space | [
"T1 Spaces",
"T1/2 Spaces"
] | [
"Definition:T1 Space",
"Definition:Topological Space",
"Definition:T1/2 Space"
] | [
"Closure of Derivative is Derivative in T1 Space",
"Definition:Set Derivative",
"Definition:Closure (Topology)",
"Topological Closure is Closed",
"Definition:Closed Set/Topology",
"Definition:T1/2 Space"
] |
proofwiki-10935 | T1/2 Space is T0 Space | Let $T = \struct {S, \tau}$ be a $T_{\frac 1 2}$ topological space.
Then $T$ is $T_0$ space. | By Characterization of T0 Space by Closures of Singletons it suffices to prove that
:$\forall x, y \in S: x \ne y \implies x \notin \set y^- \lor y \notin \set x^-$
where $\set y^-$ denotes the closure of $\set y$.
Let $x, y$ be points of $T$ such that:
:$x \ne y$
{{AimForCont}}:
:$x \in \set y^- \land y \in \set x^-$
... | Let $T = \struct {S, \tau}$ be a [[Definition:T1/2 Space|$T_{\frac 1 2}$ topological space]].
Then $T$ is [[Definition:T0 Space|$T_0$ space]]. | By [[Characterization of T0 Space by Closures of Singletons]] it suffices to prove that
:$\forall x, y \in S: x \ne y \implies x \notin \set y^- \lor y \notin \set x^-$
where $\set y^-$ denotes the [[Definition:Closure (Topology)|closure]] of $\set y$.
Let $x, y$ be [[Definition:Element|points]] of $T$ such that:
:$x ... | T1/2 Space is T0 Space | https://proofwiki.org/wiki/T1/2_Space_is_T0_Space | https://proofwiki.org/wiki/T1/2_Space_is_T0_Space | [
"T0 Spaces",
"T1/2 Spaces"
] | [
"Definition:T1/2 Space",
"Definition:T0 Space"
] | [
"Characterization of T0 Space by Closures of Singletons",
"Definition:Closure (Topology)",
"Definition:Element",
"Definition:Set Derivative",
"Definition:Open Set/Topology",
"Definition:Topological Space",
"Characterization of Derivative by Open Sets",
"Definition:Set Intersection",
"Definition:Sing... |
proofwiki-10936 | Parabolas Inscribed in Shared Tangent Lines | Let the function $\map f x = A x^2 + B x + C_1$ be a curve embedded in the Euclidean Plane.
Let $\map {y_1} x$ be the equation of the tangent line at $\tuple {Q, \map f Q}$ on $f$.
Let $\map {y_2} x$ be the equation of the tangent line at $\tuple {-Q, \map f {-Q} }$ on $f$.
Then there exists another function $\map g x$... | The tangent line at $\tuple {Q, \map f \Q}$ on $f$ is defined as:
:$\map {y_1} x = \paren {2 A Q + B} x + b_1$
where $2 A Q + B$ is the slope of the tangent line on the point $\tuple {Q, \map g Q}$ on $f$.
Substitute in the coordinates of the point $\tuple {Q, \map g Q}$ to $y_1$ and solve for $b_1$.
This will reveal t... | Let the function $\map f x = A x^2 + B x + C_1$ be a curve embedded in the [[Definition:Euclidean Plane|Euclidean Plane]].
Let $\map {y_1} x$ be the equation of the [[Definition:Tangent Line|tangent line]] at $\tuple {Q, \map f Q}$ on $f$.
Let $\map {y_2} x$ be the equation of the [[Definition:Tangent Line|tangent li... | The [[Definition:Tangent Line|tangent line]] at $\tuple {Q, \map f \Q}$ on $f$ is defined as:
:$\map {y_1} x = \paren {2 A Q + B} x + b_1$
where $2 A Q + B$ is the [[Definition:Slope of Straight Line|slope]] of the [[Definition:Tangent Line|tangent line]] on the point $\tuple {Q, \map g Q}$ on $f$.
Substitute in t... | Parabolas Inscribed in Shared Tangent Lines | https://proofwiki.org/wiki/Parabolas_Inscribed_in_Shared_Tangent_Lines | https://proofwiki.org/wiki/Parabolas_Inscribed_in_Shared_Tangent_Lines | [
"Analytic Geometry",
"Tangents"
] | [
"Definition:Euclidean Plane",
"Definition:Tangent Line",
"Definition:Tangent Line",
"Definition:Euclidean Plane",
"Definition:Tangent Line",
"Definition:Tangent Line",
"Definition:Tangent Line",
"Definition:Tangent Line"
] | [
"Definition:Tangent Line",
"Definition:Slope/Straight Line",
"Definition:Tangent Line",
"Definition:Coordinate System",
"Definition:Slope/Straight Line",
"Definition:Tangent Line",
"Definition:Coordinate System",
"Definition:Intercept/Y-Intercept",
"Definition:Intercept/Y-Intercept",
"Definition:D... |
proofwiki-10937 | Characterization of T0 Space by Closures of Singletons | Let $T = \struct {S, \tau}$ be a topological space.
Then
:$T$ is a $T_0$ space {{iff}}:
::$\forall x, y \in S: x \ne y \implies x \notin \set y^- \lor y \notin \set x^-$
where $\set y^-$ denotes the closure of $\set y$. | === Sufficient Condition ===
Let $T$ be a $T_0$ space.
Let $x, y \in S$ such that
:$x \ne y$
{{AimForCont}}
:$x \in \set y^- \land y \in \set x^-$
Then:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \set y^-
}}
{{eqn | lo= \land
| l = \set y
| o = \subseteq
| r = \set x^-
}}
{{eqn | ll= \lea... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Then
:$T$ is a [[Definition:T0 Space|$T_0$ space]] {{iff}}:
::$\forall x, y \in S: x \ne y \implies x \notin \set y^- \lor y \notin \set x^-$
where $\set y^-$ denotes the [[Definition:Closure (Topology)|closure]] of $\set y$. | === Sufficient Condition ===
Let $T$ be a [[Definition:T0 Space|$T_0$ space]].
Let $x, y \in S$ such that
:$x \ne y$
{{AimForCont}}
:$x \in \set y^- \land y \in \set x^-$
Then:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \set y^-
}}
{{eqn | lo= \land
| l = \set y
| o = \subseteq
| r =... | Characterization of T0 Space by Closures of Singletons | https://proofwiki.org/wiki/Characterization_of_T0_Space_by_Closures_of_Singletons | https://proofwiki.org/wiki/Characterization_of_T0_Space_by_Closures_of_Singletons | [
"T0 Spaces"
] | [
"Definition:Topological Space",
"Definition:T0 Space",
"Definition:Closure (Topology)"
] | [
"Definition:T0 Space",
"Topological Closure of Subset is Subset of Topological Closure",
"Closure of Topological Closure equals Closure",
"Characterization of T0 Space by Distinct Closures of Singletons",
"Proof by Contradiction",
"Characterization of T0 Space by Distinct Closures of Singletons",
"Proof... |
proofwiki-10938 | Characterization of T0 Space by Distinct Closures of Singletons | Let $T = \struct {S, \tau}$ be a topological space.
Then
:$T$ is a $T_0$ space {{iff}}
::$\forall x, y \in S: x \ne y \implies \set x^- \ne \set y^-$
where $\set y^-$ denotes the closure of $\set y$. | === Sufficient Condition ===
Let $T$ be a $T_0$ space.
Let $x, y \in S$ such that
:$x \ne y$
By definition of $T_0$ space:
:$\paren {\exists U \in \tau: x \in U \land y \notin U} \lor \paren {\exists U \in \tau: x \notin U \land y \in U}$
{{WLOG}}, suppose
:$\exists U \in \tau: x \in U \land y \notin U$
By definition o... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Then
:$T$ is a [[Definition:T0 Space|$T_0$ space]] {{iff}}
::$\forall x, y \in S: x \ne y \implies \set x^- \ne \set y^-$
where $\set y^-$ denotes the [[Definition:Closure (Topology)|closure]] of $\set y$. | === Sufficient Condition ===
Let $T$ be a [[Definition:T0 Space|$T_0$ space]].
Let $x, y \in S$ such that
:$x \ne y$
By definition of [[Definition:T0 Space|$T_0$ space]]:
:$\paren {\exists U \in \tau: x \in U \land y \notin U} \lor \paren {\exists U \in \tau: x \notin U \land y \in U}$
{{WLOG}}, suppose
:$\exists U... | Characterization of T0 Space by Distinct Closures of Singletons | https://proofwiki.org/wiki/Characterization_of_T0_Space_by_Distinct_Closures_of_Singletons | https://proofwiki.org/wiki/Characterization_of_T0_Space_by_Distinct_Closures_of_Singletons | [
"T0 Spaces"
] | [
"Definition:Topological Space",
"Definition:T0 Space",
"Definition:Closure (Topology)"
] | [
"Definition:T0 Space",
"Definition:T0 Space",
"Definition:Singleton",
"Set is Subset of its Topological Closure",
"Definition:Subset",
"Definition:Set Intersection",
"Definition:Singleton",
"Definition:Set Intersection",
"Open Set Disjoint from Set is Disjoint from Closure",
"Definition:Singleton"... |
proofwiki-10939 | Characterization of T0 Space by Closed Sets | Let $T = \struct {S, \tau}$ be a topological space.
Then
:$T$ is a $T_0$ space {{iff}}
::for every points $x, y \in S$ if $x \ne y$ then
:::there exists a closed subset $F$ of $S$ such that $x \in F$ and $y \notin F$
::or
:::there exists a closed subset $F$ of $S$ such that $x \notin F$ and $y \in F$ | === Sufficient Condition ===
Let $T$ be a $T_0$ space.
Let $x, y \in S$ such that
:$x \ne y$
By definition of $T_0$ space:
:$\paren {\exists U \in \tau: x \in U \land y \notin U} \lor \paren {\exists U \in \tau: x \notin U \land y \in U}$
{{WLOG}}, suppose:
:$\exists U \in \tau: x \in U \land y \notin U$
By definition:... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Then
:$T$ is a [[Definition:T0 Space|$T_0$ space]] {{iff}}
::for every [[Definition:Element|points]] $x, y \in S$ if $x \ne y$ then
:::there exists a [[Definition:Closed Set (Topology)|closed]] [[Definition:Subset|subset]] $F$ of $S$ ... | === Sufficient Condition ===
Let $T$ be a [[Definition:T0 Space|$T_0$ space]].
Let $x, y \in S$ such that
:$x \ne y$
By definition of [[Definition:T0 Space|$T_0$ space]]:
:$\paren {\exists U \in \tau: x \in U \land y \notin U} \lor \paren {\exists U \in \tau: x \notin U \land y \in U}$
{{WLOG}}, suppose:
:$\exists ... | Characterization of T0 Space by Closed Sets | https://proofwiki.org/wiki/Characterization_of_T0_Space_by_Closed_Sets | https://proofwiki.org/wiki/Characterization_of_T0_Space_by_Closed_Sets | [
"T0 Spaces"
] | [
"Definition:Topological Space",
"Definition:T0 Space",
"Definition:Element",
"Definition:Closed Set/Topology",
"Definition:Subset",
"Definition:Closed Set/Topology",
"Definition:Subset"
] | [
"Definition:T0 Space",
"Definition:T0 Space",
"Definition:Closed Set/Topology",
"Definition:Relative Complement",
"Definition:Relative Complement",
"Definition:Closed Set/Topology"
] |
proofwiki-10940 | Linear Eccentricity of Ellipse from Major and Minor Axis | Let $K$ be an ellipse whose major axis is $2 a$ and whose minor axis is $2 b$.
Let $c$ be the linear eccentricity of $K$.
Then:
:$a^2 = b^2 + c^2$ | :500px
Let the foci of $K$ be $F_1$ and $F_2$.
Let the vertices of $K$ be $V_1$ and $V_2$.
Let the covertices of $K$ be $C_1$ and $C_2$.
Let $P = \tuple {x, y}$ be an arbitrary point on the locus of $K$.
From the equidistance property of $K$ we have that:
:$F_1 P + F_2 P = d$
where $d$ is a constant for this particular... | Let $K$ be an [[Definition:Ellipse|ellipse]] whose [[Definition:Major Axis of Ellipse|major axis]] is $2 a$ and whose [[Definition:Minor Axis of Ellipse|minor axis]] is $2 b$.
Let $c$ be the [[Definition:Linear Eccentricity|linear eccentricity]] of $K$.
Then:
:$a^2 = b^2 + c^2$ | :[[File:EllipseFocus MajorMinorAxes.png|500px]]
Let the [[Definition:Focus of Ellipse|foci]] of $K$ be $F_1$ and $F_2$.
Let the [[Definition:Vertex of Ellipse|vertices]] of $K$ be $V_1$ and $V_2$.
Let the [[Definition:Covertex of Ellipse|covertices]] of $K$ be $C_1$ and $C_2$.
Let $P = \tuple {x, y}$ be an arbitr... | Linear Eccentricity of Ellipse from Major and Minor Axis | https://proofwiki.org/wiki/Linear_Eccentricity_of_Ellipse_from_Major_and_Minor_Axis | https://proofwiki.org/wiki/Linear_Eccentricity_of_Ellipse_from_Major_and_Minor_Axis | [
"Linear Eccentricity of Ellipse from Major and Minor Axis",
"Linear Eccentricity",
"Major Axis of Ellipse",
"Minor Axis of Ellipse",
"Ellipses"
] | [
"Definition:Ellipse",
"Definition:Ellipse/Major Axis",
"Definition:Ellipse/Minor Axis",
"Definition:Linear Eccentricity"
] | [
"File:EllipseFocus MajorMinorAxes.png",
"Definition:Ellipse/Focus",
"Definition:Ellipse/Vertex",
"Definition:Ellipse/Covertex",
"Definition:Point",
"Definition:Locus",
"Definition:Ellipse/Equidistance",
"Definition:Constant",
"Definition:Ellipse",
"Definition:Point",
"Pythagoras's Theorem",
"P... |
proofwiki-10941 | Equidistance of Ellipse equals Major Axis | Let $K$ be an ellipse whose foci are $F_1$ and $F_2$.
Let $P$ be an arbitrary point on $K$.
Let $d$ be the constant distance such that:
:$d_1 + d_2 = d$
where:
:$d_1 = P F_1$
:$d_2 = P F_2$
Then $d$ is equal to the major axis of $K$. | :400px
By the equidistance property of $K$:
:$d_1 + d_2 = d$
applies to all points $P$ on $K$.
Thus it also applies to the two vertices $V_1$ and $V_2$:
:$V_1 F_1 + V_1 F_2 = d$
:$V_2 F_1 + V_2 F_2 = d$
Adding:
:$V_1 F_1 + V_2 F_1 + V_1 F_2 + V_2 F_2 = 2 d$
But:
:$V_1 F_1 + V_2 F_1 = V_1 V_2$
:$V_1 F_2 + V_2 F_2 = V_1 ... | Let $K$ be an [[Definition:Ellipse|ellipse]] whose [[Definition:Focus of Ellipse|foci]] are $F_1$ and $F_2$.
Let $P$ be an arbitrary [[Definition:Point|point]] on $K$.
Let $d$ be the [[Definition:Constant|constant]] [[Definition:Distance between Points|distance]] such that:
:$d_1 + d_2 = d$
where:
:$d_1 = P F_1$
:$d... | :[[File:EllipseEquidistanceMajorAxis.png|400px]]
By the [[Definition:Equidistance Property of Ellipse|equidistance property]] of $K$:
:$d_1 + d_2 = d$
applies to all [[Definition:Point|points]] $P$ on $K$.
Thus it also applies to the two [[Definition:Vertex of Ellipse|vertices]] $V_1$ and $V_2$:
:$V_1 F_1 + V_1 F_2... | Equidistance of Ellipse equals Major Axis | https://proofwiki.org/wiki/Equidistance_of_Ellipse_equals_Major_Axis | https://proofwiki.org/wiki/Equidistance_of_Ellipse_equals_Major_Axis | [
"Ellipses"
] | [
"Definition:Ellipse",
"Definition:Ellipse/Focus",
"Definition:Point",
"Definition:Constant",
"Definition:Distance between Points",
"Definition:Ellipse/Major Axis"
] | [
"File:EllipseEquidistanceMajorAxis.png",
"Definition:Ellipse/Equidistance",
"Definition:Point",
"Definition:Ellipse/Vertex",
"Definition:Ellipse/Major Axis"
] |
proofwiki-10942 | Equidistance of Hyperbola equals Transverse Axis | Let $K$ be an hyperbola whose foci are $F_1$ and $F_2$.
Let $P$ be an arbitrary point on $K$.
Let $d$ be the constant distance such that:
:$\size {d_1 - d_2} = d$
where:
:$d_1 = P F_1$
:$d_2 = P F_2$
Then $d$ is equal to the transverse axis of $K$. |
:400px
By the equidistance property of $K$:
:$\size {d_1 - d_2} = d$
applies to all points $P$ on $K$.
Thus it also applies to the two vertices $V_1$ and $V_2$.
Observing the signs of $\size {d_1 - d_2}$ as appropriate:
:$V_1 F_2 - V_1 F_1 = d$
:$V_2 F_1 - V_2 F_2 = d$
Adding:
:$\paren {V_1 F_2 - V_2 F_2} + \par... | Let $K$ be an [[Definition:Hyperbola|hyperbola]] whose [[Definition:Focus of Hyperbola|foci]] are $F_1$ and $F_2$.
Let $P$ be an arbitrary [[Definition:Point|point]] on $K$.
Let $d$ be the [[Definition:Constant|constant]] [[Definition:Distance between Points|distance]] such that:
:$\size {d_1 - d_2} = d$
where:
:$d_... |
:[[File:HyperbolaEquidistanceTransverseAxis.png|400px]]
By the [[Definition:Equidistance Property of Hyperbola|equidistance property]] of $K$:
:$\size {d_1 - d_2} = d$
applies to all [[Definition:Point|points]] $P$ on $K$.
Thus it also applies to the two [[Definition:Vertex of Hyperbola|vertices]] $V_1$ and... | Equidistance of Hyperbola equals Transverse Axis | https://proofwiki.org/wiki/Equidistance_of_Hyperbola_equals_Transverse_Axis | https://proofwiki.org/wiki/Equidistance_of_Hyperbola_equals_Transverse_Axis | [
"Hyperbolas"
] | [
"Definition:Hyperbola",
"Definition:Hyperbola/Focus",
"Definition:Point",
"Definition:Constant",
"Definition:Distance between Points",
"Definition:Hyperbola/Transverse Axis"
] | [
"File:HyperbolaEquidistanceTransverseAxis.png",
"Definition:Hyperbola/Equidistance",
"Definition:Point",
"Definition:Hyperbola/Vertex",
"Definition:Hyperbola/Transverse Axis",
"Category:Hyperbolas"
] |
proofwiki-10943 | Equivalence of Definitions of Ellipse | The following definitions of an ellipse are equivalent: | Let $K$ be an ellipse aligned in a cartesian plane in reduced form.
Thus its foci are at $\tuple {\mathop \pm c, 0}$.
Let:
:the major axis of $K$ have length $2 a$
:the minor axis of $K$ have length $2 b$.
From Equation of Ellipse in Reduced Form, the equation of $K$ is:
:$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$
T... | The following definitions of an [[Definition:Ellipse|ellipse]] are [[Definition:Logical Equivalence|equivalent]]: | Let $K$ be an [[Definition:Ellipse|ellipse]] aligned in a [[Definition:Cartesian Plane|cartesian plane]] in [[Definition:Reduced Form of Ellipse|reduced form]].
Thus its [[Definition:Focus of Ellipse|foci]] are at $\tuple {\mathop \pm c, 0}$.
Let:
:the [[Definition:Major Axis of Ellipse|major axis]] of $K$ have [[De... | Equivalence of Definitions of Ellipse | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Ellipse | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Ellipse | [
"Ellipses"
] | [
"Definition:Ellipse",
"Definition:Logical Equivalence"
] | [
"Definition:Ellipse",
"Definition:Cartesian Plane",
"Definition:Conic Section/Reduced Form/Ellipse",
"Definition:Ellipse/Focus",
"Definition:Ellipse/Major Axis",
"Definition:Linear Measure/Length",
"Definition:Ellipse/Minor Axis",
"Definition:Linear Measure/Length",
"Equation of Ellipse in Reduced F... |
proofwiki-10944 | Focus of Hyperbola from Transverse and Conjugate Axis | Let $K$ be a hyperbola whose transverse axis is $2 a$ and whose conjugate axis is $2 b$.
Let $c$ be the distance of the foci of $K$ from the center.
Then:
:$c^2 = a^2 + b^2$ |
:500px
Let the foci of $K$ be $F_1$ and $F_2$.
Let the vertices of $K$ be $V_1$ and $V_2$.
Let the covertices of $K$ be $C_1$ and $C_2$.
Let $P = \tuple {x, y}$ be an arbitrary point on the locus of $K$.
From the equidistance property of $K$ we have that:
:$\size {F_1 P - F_2 P} = d$
where $d$ is a constant for ... | Let $K$ be a [[Definition:Hyperbola|hyperbola]] whose [[Definition:Transverse Axis of Hyperbola|transverse axis]] is $2 a$ and whose [[Definition:Conjugate Axis of Hyperbola|conjugate axis]] is $2 b$.
Let $c$ be the [[Definition:Distance between Points|distance]] of the [[Definition:Focus of Hyperbola|foci]] of $K$ fr... |
:[[File:HyperbolaFocusTransConj.png|500px]]
Let the [[Definition:Focus of Hyperbola|foci]] of $K$ be $F_1$ and $F_2$.
Let the [[Definition:Vertex of Hyperbola|vertices]] of $K$ be $V_1$ and $V_2$.
Let the [[Definition:Covertex of Hyperbola|covertices]] of $K$ be $C_1$ and $C_2$.
Let $P = \tuple {x, y}$ be... | Focus of Hyperbola from Transverse and Conjugate Axis | https://proofwiki.org/wiki/Focus_of_Hyperbola_from_Transverse_and_Conjugate_Axis | https://proofwiki.org/wiki/Focus_of_Hyperbola_from_Transverse_and_Conjugate_Axis | [
"Hyperbolas"
] | [
"Definition:Hyperbola",
"Definition:Hyperbola/Transverse Axis",
"Definition:Hyperbola/Conjugate Axis",
"Definition:Distance between Points",
"Definition:Hyperbola/Focus",
"Definition:Ellipse/Center"
] | [
"File:HyperbolaFocusTransConj.png",
"Definition:Hyperbola/Focus",
"Definition:Hyperbola/Vertex",
"Definition:Hyperbola/Covertex",
"Definition:Point",
"Definition:Locus",
"Definition:Hyperbola/Equidistance",
"Definition:Constant",
"Definition:Hyperbola",
"Definition:Point",
"Category:Hyperbolas"
... |
proofwiki-10945 | Equation of Hyperbola in Reduced Form/Cartesian Frame/Parametric Form 1 | The right-hand branch of $K$ can be expressed in parametric form as:
:$\begin {cases} x = a \cosh \theta \\ y = b \sinh \theta \end {cases}$ | Let the point $\tuple {x, y}$ satisfy the equations:
{{begin-eqn}}
{{eqn | l = x
| r = a \cosh \theta
}}
{{eqn | l = y
| r = b \sinh \theta
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \frac {x^2} {a^2} - \frac {y^2} {b^2}
| r = \frac {\paren {a \cosh \theta}^2} {a^2} - \frac {\paren {b \sinh \theta... | The right-hand [[Definition:Branch of Hyperbola|branch]] of $K$ can be expressed in [[Definition:Parametric Equation|parametric form]] as:
:$\begin {cases} x = a \cosh \theta \\ y = b \sinh \theta \end {cases}$ | Let the point $\tuple {x, y}$ satisfy the equations:
{{begin-eqn}}
{{eqn | l = x
| r = a \cosh \theta
}}
{{eqn | l = y
| r = b \sinh \theta
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \frac {x^2} {a^2} - \frac {y^2} {b^2}
| r = \frac {\paren {a \cosh \theta}^2} {a^2} - \frac {\paren {b \sinh \t... | Equation of Hyperbola in Reduced Form/Cartesian Frame/Parametric Form 1 | https://proofwiki.org/wiki/Equation_of_Hyperbola_in_Reduced_Form/Cartesian_Frame/Parametric_Form_1 | https://proofwiki.org/wiki/Equation_of_Hyperbola_in_Reduced_Form/Cartesian_Frame/Parametric_Form_1 | [
"Equation of Hyperbola in Reduced Form"
] | [
"Definition:Hyperbola/Branch",
"Definition:Parametric Equation"
] | [
"Difference of Squares of Hyperbolic Cosine and Sine"
] |
proofwiki-10946 | Set of Condensation Points is Subset of Derivative | Let $T = \left({S, \tau}\right)$ be a topological space.
Let $A$ be a subset of $S$.
Then:
:$A^0 \subseteq A'$
where
:$A^0$ denotes the set of condensation points of $A$
:$A'$ denotes the derivative of $A$ | Let $x \in A^0$.
By definition of set of condensation points:
:$x$ is condensation point of $A$
By definition of condensation point:
:$x$ is limit point of $A$
By definition of derived set:
:$x \in A'$
{{qed}} | Let $T = \left({S, \tau}\right)$ be a [[Definition:Topological Space|topological space]].
Let $A$ be a [[Definition:Subset|subset]] of $S$.
Then:
:$A^0 \subseteq A'$
where
:$A^0$ denotes the [[Definition:Set of Condensation Points|set of condensation points]] of $A$
:$A'$ denotes the [[Definition:Set Derivative|deri... | Let $x \in A^0$.
By definition of [[Definition:Set of Condensation Points|set of condensation points]]:
:$x$ is [[Definition:Condensation Point|condensation point]] of $A$
By definition of [[Definition:Condensation Point|condensation point]]:
:$x$ is [[Definition:Limit Point of Set|limit point]] of $A$
By definition... | Set of Condensation Points is Subset of Derivative | https://proofwiki.org/wiki/Set_of_Condensation_Points_is_Subset_of_Derivative | https://proofwiki.org/wiki/Set_of_Condensation_Points_is_Subset_of_Derivative | [
"Condensation Points",
"Set Derivatives"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Set of Condensation Points",
"Definition:Set Derivative"
] | [
"Definition:Set of Condensation Points",
"Definition:Condensation Point",
"Definition:Condensation Point",
"Definition:Limit Point/Topology/Set",
"Definition:Derived Set"
] |
proofwiki-10947 | Closure of Set of Condensation Points equals Itself | Let $T = \struct {S, \tau}$ be a topological space.
Let $A$ be a subset of $S$.
Then:
:$\paren {A^0}^- = A^0$
where
:$A^0$ denotes the set of condensation points of $A$
:$A^-$ denotes the closure of $A$ | By Set is Subset of its Topological Closure:
:$A^0 \subseteq \paren {A^0}^-$
To prove the equality by definition of set equality it suffices to show the inclusion:
:$\paren {A^0}^- \subseteq A^0$
Let $x \in \paren {A^0}^-$.
We will prove that
:$(1): \quad \forall U \in \tau: x \in U \implies A \cap U$ is uncountable
Le... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A$ be a [[Definition:Subset|subset]] of $S$.
Then:
:$\paren {A^0}^- = A^0$
where
:$A^0$ denotes the [[Definition:Set of Condensation Points|set of condensation points]] of $A$
:$A^-$ denotes the [[Definition:Closure (Topology)|... | By [[Set is Subset of its Topological Closure]]:
:$A^0 \subseteq \paren {A^0}^-$
To prove the equality by definition of [[Definition:Set Equality|set equality]] it suffices to show the inclusion:
:$\paren {A^0}^- \subseteq A^0$
Let $x \in \paren {A^0}^-$.
We will prove that
:$(1): \quad \forall U \in \tau: x \in U \... | Closure of Set of Condensation Points equals Itself | https://proofwiki.org/wiki/Closure_of_Set_of_Condensation_Points_equals_Itself | https://proofwiki.org/wiki/Closure_of_Set_of_Condensation_Points_equals_Itself | [
"Condensation Points",
"Set Closures"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Set of Condensation Points",
"Definition:Closure (Topology)"
] | [
"Set is Subset of its Topological Closure",
"Definition:Set Equality",
"Definition:Countable Set",
"Definition:Open Set/Topology",
"Definition:Subset",
"Condition for Point being in Closure",
"Definition:Empty Set",
"Definition:Set Intersection",
"Definition:Set of Condensation Points",
"Definitio... |
proofwiki-10948 | First Order ODE/(x + y) dx = (x - y) dy | is a homogeneous differential equation with general solution:
:$\arctan \dfrac y x = \ln \sqrt {x^2 + y^2} + C$ | Let:
:$\map M {x, y} = x + y$
:$\map N {x, y} = x - y$
We have that:
:$\map M {t x, t y} = t x + t y = t \paren {x + y} = t \map M {x, y}$
:$\map N {t x, t y} = t x - t y = t \paren {x - y} = t \map N {x, y}$
Thus both $M$ and $N$ are homogeneous functions of degree $1$.
Thus by definition $(1)$ is a homogeneous di... | is a [[Definition:Homogeneous Differential Equation|homogeneous differential equation]] with [[Definition:General Solution to Differential Equation|general solution]]:
:$\arctan \dfrac y x = \ln \sqrt {x^2 + y^2} + C$ | Let:
:$\map M {x, y} = x + y$
:$\map N {x, y} = x - y$
We have that:
:$\map M {t x, t y} = t x + t y = t \paren {x + y} = t \map M {x, y}$
:$\map N {t x, t y} = t x - t y = t \paren {x - y} = t \map N {x, y}$
Thus both $M$ and $N$ are [[Definition:Homogeneous Real Function|homogeneous functions]] of [[Definition... | First Order ODE/(x + y) dx = (x - y) dy/Proof 1 | https://proofwiki.org/wiki/First_Order_ODE/(x_+_y)_dx_=_(x_-_y)_dy | https://proofwiki.org/wiki/First_Order_ODE/(x_+_y)_dx_=_(x_-_y)_dy/Proof_1 | [
"Examples of First Order ODEs",
"Examples of Homogeneous Differential Equation",
"First Order ODE/(x + y) dx = (x - y) dy"
] | [
"Definition:Homogeneous Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Function/Real Space",
"Definition:Homogeneous Function/Real Space/Degree",
"Definition:Homogeneous Differential Equation",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Solution to Homogeneous Differential Equation",
"Primitive of Reciprocal of x squared pl... |
proofwiki-10949 | First Order ODE/(x + y) dx = (x - y) dy | is a homogeneous differential equation with general solution:
:$\arctan \dfrac y x = \ln \sqrt {x^2 + y^2} + C$ | We have:
{{begin-eqn}}
{{eqn | l = \paren {x + y} \rd x
| r = \paren {x - y} \rd y
| c =
}}
{{eqn | ll= \leadsto
| l = x \rd y - y \rd x
| r = x \rd x + y \rd y
| c = rearranging
}}
{{eqn | ll= \leadsto
| l = \frac {x \rd y - y \rd x} {x^2 + y^2}
| r = \frac {x \rd x + y \rd y... | is a [[Definition:Homogeneous Differential Equation|homogeneous differential equation]] with [[Definition:General Solution to Differential Equation|general solution]]:
:$\arctan \dfrac y x = \ln \sqrt {x^2 + y^2} + C$ | We have:
{{begin-eqn}}
{{eqn | l = \paren {x + y} \rd x
| r = \paren {x - y} \rd y
| c =
}}
{{eqn | ll= \leadsto
| l = x \rd y - y \rd x
| r = x \rd x + y \rd y
| c = rearranging
}}
{{eqn | ll= \leadsto
| l = \frac {x \rd y - y \rd x} {x^2 + y^2}
| r = \frac {x \rd x + y \rd y... | First Order ODE/(x + y) dx = (x - y) dy/Proof 2 | https://proofwiki.org/wiki/First_Order_ODE/(x_+_y)_dx_=_(x_-_y)_dy | https://proofwiki.org/wiki/First_Order_ODE/(x_+_y)_dx_=_(x_-_y)_dy/Proof_2 | [
"Examples of First Order ODEs",
"Examples of Homogeneous Differential Equation",
"First Order ODE/(x + y) dx = (x - y) dy"
] | [
"Definition:Homogeneous Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Differential of Sum of Squares",
"Differential of Arctangent of Quotient"
] |
proofwiki-10950 | First Order ODE/(x^2 - 2 y^2) dx + x y dy = 0 | is a homogeneous differential equation with solution:
:$y^2 = x^2 + C x^4$ | $(1)$ can also be rendered:
:$\dfrac {\d y} {\d x} = -\dfrac {x^2 - 2 y^2} {x y}$
Let:
:$\map M {x, y} = x^2 - 2 y^2$
:$\map N {x, y} = x y$
Put $t x, t y$ for $x, y$:
{{begin-eqn}}
{{eqn | l = \map M {t x, t y}
| r = \paren {t x}^2 - 2 \paren {t y}^2
| c =
}}
{{eqn | r = t^2 \paren {x^2 - 2 y^2}
| c... | is a [[Definition:Homogeneous Differential Equation|homogeneous differential equation]] with [[Definition:General Solution to Differential Equation|solution]]:
:$y^2 = x^2 + C x^4$ | $(1)$ can also be rendered:
:$\dfrac {\d y} {\d x} = -\dfrac {x^2 - 2 y^2} {x y}$
Let:
:$\map M {x, y} = x^2 - 2 y^2$
:$\map N {x, y} = x y$
Put $t x, t y$ for $x, y$:
{{begin-eqn}}
{{eqn | l = \map M {t x, t y}
| r = \paren {t x}^2 - 2 \paren {t y}^2
| c =
}}
{{eqn | r = t^2 \paren {x^2 - 2 y^2}
... | First Order ODE/(x^2 - 2 y^2) dx + x y dy = 0 | https://proofwiki.org/wiki/First_Order_ODE/(x^2_-_2_y^2)_dx_+_x_y_dy_=_0 | https://proofwiki.org/wiki/First_Order_ODE/(x^2_-_2_y^2)_dx_+_x_y_dy_=_0 | [
"Examples of Homogeneous Differential Equation",
"Examples of First Order ODEs"
] | [
"Definition:Homogeneous Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Function/Real Space",
"Definition:Homogeneous Function/Real Space/Degree",
"Definition:Homogeneous Differential Equation",
"Solution to Homogeneous Differential Equation",
"Primitive of x over x squared minus a squared"
] |
proofwiki-10951 | First Order ODE/x^2 y' - 3 x y - 2 y^2 = 0 | is a homogeneous differential equation with solution:
:$y = C x^2 \paren {x + y}$ | Let:
:$\map M {x, y} = 3 x y + 2 y^2$
:$\map N {x, y} = x^2$
Put $t x, t y$ for $x, y$:
{{begin-eqn}}
{{eqn | l = \map M {t x, t y}
| r = 3 t x t y + 2 \paren {t y}^2
| c =
}}
{{eqn | r = t^2 \paren {3 x y + 2 y^2}
| c =
}}
{{eqn | r = t^2 \, \map M {x, y}
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{... | is a [[Definition:Homogeneous Differential Equation|homogeneous differential equation]] with [[Definition:General Solution to Differential Equation|solution]]:
:$y = C x^2 \paren {x + y}$ | Let:
:$\map M {x, y} = 3 x y + 2 y^2$
:$\map N {x, y} = x^2$
Put $t x, t y$ for $x, y$:
{{begin-eqn}}
{{eqn | l = \map M {t x, t y}
| r = 3 t x t y + 2 \paren {t y}^2
| c =
}}
{{eqn | r = t^2 \paren {3 x y + 2 y^2}
| c =
}}
{{eqn | r = t^2 \, \map M {x, y}
| c =
}}
{{end-eqn}}
{{begin-eqn... | First Order ODE/x^2 y' - 3 x y - 2 y^2 = 0 | https://proofwiki.org/wiki/First_Order_ODE/x^2_y'_-_3_x_y_-_2_y^2_=_0 | https://proofwiki.org/wiki/First_Order_ODE/x^2_y'_-_3_x_y_-_2_y^2_=_0 | [
"Examples of Homogeneous Differential Equation",
"Examples of First Order ODEs"
] | [
"Definition:Homogeneous Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Function/Real Space",
"Definition:Homogeneous Function/Real Space/Degree",
"Definition:Homogeneous Differential Equation",
"Solution to Homogeneous Differential Equation",
"Primitive of Reciprocal of x by a x + b"
] |
proofwiki-10952 | Set of Condensation Points is Monotone | Let $T = \struct {S, \tau}$ be a topological space.
Let $A, B$ be subsets of $S$.
Then:
:$A \subseteq B \implies {A^0} \subseteq B^0$
where
:$A^0$ denotes the set of condensation points of $A$ | Assume
:$A \subseteq B$
Let $x \in A^0$.
By definition of set of condensation points:
:$x$ is condensation point of $A$
By definition of condensation point:
:$x$ is limit point of $A$ such that $\forall U \in \tau: A \cap U$ is uncountable
Thus by Limit Point of Subset is Limit Point of Set:
:$x$ is limit point of $B$
... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A, B$ be [[Definition:Subset|subsets]] of $S$.
Then:
:$A \subseteq B \implies {A^0} \subseteq B^0$
where
:$A^0$ denotes the [[Definition:Set of Condensation Points|set of condensation points]] of $A$ | Assume
:$A \subseteq B$
Let $x \in A^0$.
By definition of [[Definition:Set of Condensation Points|set of condensation points]]:
:$x$ is [[Definition:Condensation Point|condensation point]] of $A$
By definition of [[Definition:Condensation Point|condensation point]]:
:$x$ is [[Definition:Limit Point of Set|limit poin... | Set of Condensation Points is Monotone | https://proofwiki.org/wiki/Set_of_Condensation_Points_is_Monotone | https://proofwiki.org/wiki/Set_of_Condensation_Points_is_Monotone | [
"Condensation Points"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Set of Condensation Points"
] | [
"Definition:Set of Condensation Points",
"Definition:Condensation Point",
"Definition:Condensation Point",
"Definition:Limit Point/Topology/Set",
"Definition:Countable Set",
"Limit Point of Subset is Limit Point of Set",
"Definition:Limit Point/Topology/Set",
"Definition:Condensation Point",
"Defini... |
proofwiki-10953 | First Order ODE/x^2 y' = 3 (x^2 + y^2) arctan (y over x) + x y | is a homogeneous differential equation with solution:
:$y = x \tan C x^3$ | Let:
:$\map M {x, y} = 3 \paren {x^2 + y^2} \arctan \dfrac y x + x y$
:$\map N {x, y} = x^2$
Put $t x, t y$ for $x, y$:
{{begin-eqn}}
{{eqn | l = \map M {t x, t y}
| r = 3 \paren {\paren {t x}^2 + \paren {t y}^2} \arctan \dfrac {t y} {t x} + t x t y
| c =
}}
{{eqn | r = t^2 \paren {3 \paren {x^2 + y^2} \ar... | is a [[Definition:Homogeneous Differential Equation|homogeneous differential equation]] with [[Definition:General Solution to Differential Equation|solution]]:
:$y = x \tan C x^3$ | Let:
:$\map M {x, y} = 3 \paren {x^2 + y^2} \arctan \dfrac y x + x y$
:$\map N {x, y} = x^2$
Put $t x, t y$ for $x, y$:
{{begin-eqn}}
{{eqn | l = \map M {t x, t y}
| r = 3 \paren {\paren {t x}^2 + \paren {t y}^2} \arctan \dfrac {t y} {t x} + t x t y
| c =
}}
{{eqn | r = t^2 \paren {3 \paren {x^2 + y^2} ... | First Order ODE/x^2 y' = 3 (x^2 + y^2) arctan (y over x) + x y | https://proofwiki.org/wiki/First_Order_ODE/x^2_y'_=_3_(x^2_+_y^2)_arctan_(y_over_x)_+_x_y | https://proofwiki.org/wiki/First_Order_ODE/x^2_y'_=_3_(x^2_+_y^2)_arctan_(y_over_x)_+_x_y | [
"Examples of Homogeneous Differential Equation",
"Examples of First Order ODEs"
] | [
"Definition:Homogeneous Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Function/Real Space",
"Definition:Homogeneous Function/Real Space/Degree",
"Definition:Homogeneous Differential Equation",
"Solution to Homogeneous Differential Equation",
"Derivative of Arctangent Function",
"Primitive of Reciprocal"
] |
proofwiki-10954 | First Order ODE/x sine (y over x) y' = y sine (y over x) + x | is a homogeneous differential equation with solution:
:$\cos \dfrac y x + \ln C x = 0$ | Let:
:$\map M {x, y} = y \sin \dfrac y x + x$
:$\map N {x, y} = x \sin \dfrac y x$
Put $t x, t y$ for $x, y$:
{{begin-eqn}}
{{eqn | l = \map M {t x, t y}
| r = t y \sin \dfrac t y t x + t x
| c =
}}
{{eqn | r = t \paren {y \sin \dfrac y x + x}
| c =
}}
{{eqn | r = t \, \map M {x, y}
| c =
}}
... | is a [[Definition:Homogeneous Differential Equation|homogeneous differential equation]] with [[Definition:General Solution to Differential Equation|solution]]:
:$\cos \dfrac y x + \ln C x = 0$ | Let:
:$\map M {x, y} = y \sin \dfrac y x + x$
:$\map N {x, y} = x \sin \dfrac y x$
Put $t x, t y$ for $x, y$:
{{begin-eqn}}
{{eqn | l = \map M {t x, t y}
| r = t y \sin \dfrac t y t x + t x
| c =
}}
{{eqn | r = t \paren {y \sin \dfrac y x + x}
| c =
}}
{{eqn | r = t \, \map M {x, y}
| c =
... | First Order ODE/x sine (y over x) y' = y sine (y over x) + x | https://proofwiki.org/wiki/First_Order_ODE/x_sine_(y_over_x)_y'_=_y_sine_(y_over_x)_+_x | https://proofwiki.org/wiki/First_Order_ODE/x_sine_(y_over_x)_y'_=_y_sine_(y_over_x)_+_x | [
"Examples of Homogeneous Differential Equation",
"Examples of First Order ODEs"
] | [
"Definition:Homogeneous Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Function/Real Space",
"Definition:Homogeneous Function/Real Space/Degree",
"Definition:Homogeneous Differential Equation",
"Solution to Homogeneous Differential Equation",
"Primitive of Sine Function"
] |
proofwiki-10955 | First Order ODE/x y' = y + 2 x exp (- y over x) | is a homogeneous differential equation with solution:
:$e^{y / x} = \ln x^2 + C$ | Let:
:$\map M {x, y} = y + 2 x e^{-y/x}$
:$\map N {x, y} = x$
Put $t x, t y$ for $x, y$:
{{begin-eqn}}
{{eqn | l = \map M {t x, t y}
| r = t y + 2 t x e^{-t y / t x}
| c =
}}
{{eqn | r = t \paren {y + 2 x e^{-y / x} }
| c =
}}
{{eqn | r = t \, \map M {x, y}
| c =
}}
{{end-eqn}}
{{begin-eqn}}
... | is a [[Definition:Homogeneous Differential Equation|homogeneous differential equation]] with [[Definition:General Solution to Differential Equation|solution]]:
:$e^{y / x} = \ln x^2 + C$ | Let:
:$\map M {x, y} = y + 2 x e^{-y/x}$
:$\map N {x, y} = x$
Put $t x, t y$ for $x, y$:
{{begin-eqn}}
{{eqn | l = \map M {t x, t y}
| r = t y + 2 t x e^{-t y / t x}
| c =
}}
{{eqn | r = t \paren {y + 2 x e^{-y / x} }
| c =
}}
{{eqn | r = t \, \map M {x, y}
| c =
}}
{{end-eqn}}
{{begin-eq... | First Order ODE/x y' = y + 2 x exp (- y over x) | https://proofwiki.org/wiki/First_Order_ODE/x_y'_=_y_+_2_x_exp_(-_y_over_x) | https://proofwiki.org/wiki/First_Order_ODE/x_y'_=_y_+_2_x_exp_(-_y_over_x) | [
"Examples of Homogeneous Differential Equation",
"Examples of First Order ODEs"
] | [
"Definition:Homogeneous Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Function/Real Space",
"Definition:Homogeneous Function/Real Space/Degree",
"Definition:Homogeneous Differential Equation",
"Solution to Homogeneous Differential Equation",
"Primitive of Exponential Function"
] |
proofwiki-10956 | First Order ODE in form y' = f (a x + b y + c) | The first order ODE:
:$\dfrac {\d y} {\d x} = \map f {a x + b y + c}$
can be solved by substituting:
:$z := a x + b y + c$
to obtain:
:$\ds x = \int \frac {\d z} {b \map f z + a}$ | We have:
:$\dfrac {\d y} {\d x} = \map f {a x + b y + c}$
Put:
: $z := a x + b y + c$
Then:
{{begin-eqn}}
{{eqn | l = z
| r = a x + b y + c
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = a + b \dfrac {\d y} {\d x}
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d y} {\d... | The [[Definition:First Order Ordinary Differential Equation|first order ODE]]:
:$\dfrac {\d y} {\d x} = \map f {a x + b y + c}$
can be solved by substituting:
:$z := a x + b y + c$
to obtain:
:$\ds x = \int \frac {\d z} {b \map f z + a}$ | We have:
:$\dfrac {\d y} {\d x} = \map f {a x + b y + c}$
Put:
: $z := a x + b y + c$
Then:
{{begin-eqn}}
{{eqn | l = z
| r = a x + b y + c
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = a + b \dfrac {\d y} {\d x}
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d y} {... | First Order ODE in form y' = f (a x + b y + c) | https://proofwiki.org/wiki/First_Order_ODE_in_form_y'_=_f_(a_x_+_b_y_+_c) | https://proofwiki.org/wiki/First_Order_ODE_in_form_y'_=_f_(a_x_+_b_y_+_c) | [
"First Order ODEs"
] | [
"Definition:First Order Ordinary Differential Equation"
] | [
"Solution to Separable Differential Equation"
] |
proofwiki-10957 | First Order ODE/y' = (x + y)^2 | The first order ODE:
:$\dfrac {\d y} {\d x} = \paren {x + y}^2$
has the general solution:
:$x + y = \map \tan {x + C}$ | Make the substitution:
:$z = x + y$
Then from First Order ODE in form $y' = f (a x + b y + c)$ with $a = b = 1$:
{{begin-eqn}}
{{eqn | l = x
| r = \int \frac {\d z} {z^2 + 1}
| c =
}}
{{eqn | r = \arctan z + C_1
| c = Primitive of $\dfrac 1 {x^2 + a^2}$
}}
{{eqn | ll= \leadsto
| l = y - C_1
... | The [[Definition:First Order ODE|first order ODE]]:
:$\dfrac {\d y} {\d x} = \paren {x + y}^2$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$x + y = \map \tan {x + C}$ | Make the substitution:
:$z = x + y$
Then from [[First Order ODE in form y' = f (a x + b y + c)|First Order ODE in form $y' = f (a x + b y + c)$]] with $a = b = 1$:
{{begin-eqn}}
{{eqn | l = x
| r = \int \frac {\d z} {z^2 + 1}
| c =
}}
{{eqn | r = \arctan z + C_1
| c = [[Primitive of Reciprocal of x... | First Order ODE/y' = (x + y)^2 | https://proofwiki.org/wiki/First_Order_ODE/y'_=_(x_+_y)^2 | https://proofwiki.org/wiki/First_Order_ODE/y'_=_(x_+_y)^2 | [
"Examples of First Order ODEs"
] | [
"Definition:First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"First Order ODE in form y' = f (a x + b y + c)",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form"
] |
proofwiki-10958 | First Order ODE/y' = sin^2 (x - y + 1) | The first order ODE:
:$\dfrac {\d y} {\d x} = \map {\sin^2} {x - y + 1}^2$
has the general solution:
:$\map \tan {x - y + 1} = x + C$ | Make the substitution:
:$z = x - y + 1$
Then from First Order ODE in form $y' = f (a x + b y + c)$ with $a = 1, b = - 1$:
{{begin-eqn}}
{{eqn | l = x
| r = \int \frac {\d z} {- \sin^2 z + 1}
| c =
}}
{{eqn | r = \int \frac {\d z} {\cos^2 z}
| c = Sum of Squares of Sine and Cosine
}}
{{eqn | r = \int ... | The [[Definition:First Order ODE|first order ODE]]:
:$\dfrac {\d y} {\d x} = \map {\sin^2} {x - y + 1}^2$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$\map \tan {x - y + 1} = x + C$ | Make the substitution:
:$z = x - y + 1$
Then from [[First Order ODE in form y' = f (a x + b y + c)|First Order ODE in form $y' = f (a x + b y + c)$]] with $a = 1, b = - 1$:
{{begin-eqn}}
{{eqn | l = x
| r = \int \frac {\d z} {- \sin^2 z + 1}
| c =
}}
{{eqn | r = \int \frac {\d z} {\cos^2 z}
| c = [... | First Order ODE/y' = sin^2 (x - y + 1) | https://proofwiki.org/wiki/First_Order_ODE/y'_=_sin^2_(x_-_y_+_1) | https://proofwiki.org/wiki/First_Order_ODE/y'_=_sin^2_(x_-_y_+_1) | [
"Examples of First Order ODEs"
] | [
"Definition:First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"First Order ODE in form y' = f (a x + b y + c)",
"Sum of Squares of Sine and Cosine",
"Secant is Reciprocal of Cosine",
"Primitive of Square of Secant Function"
] |
proofwiki-10959 | First Order ODE in form y' = F ((a x + b y + c) over (d x + e y + f)) | The first order ODE:
:$\dfrac {\d y} {\d x} = \map F {\dfrac {a x + b y + c} {d x + e y + f} }$
such that:
:$ a e \ne b d$
can be solved by substituting:
:$x := z - h$
:$y := w - k$
where:
:$h = \dfrac {c e - b f} {a e - b d}$
:$k = \dfrac {a f - c d} {a e - b d}$
to obtain:
:$\dfrac {\d w} {\d z} = \map F {\dfrac {a z... | We have:
:$\dfrac {\d y} {\d x} = \map F {\dfrac {a x + b y + c} {d x + e y + f} }$
Make the substitutions:
:$x := z - h$
:$y := w - k$
We have:
:$\dfrac {\d x} {\d z} = 1$
:$\dfrac {\d y} {\d w} = 1$
Thus:
{{begin-eqn}}
{{eqn | l = \frac {\d w} {\d z}
| r = \map F {\frac {a \paren {z - h} + b \paren {w - k} + c}... | The [[Definition:First Order Ordinary Differential Equation|first order ODE]]:
:$\dfrac {\d y} {\d x} = \map F {\dfrac {a x + b y + c} {d x + e y + f} }$
such that:
:$ a e \ne b d$
can be solved by substituting:
:$x := z - h$
:$y := w - k$
where:
:$h = \dfrac {c e - b f} {a e - b d}$
:$k = \dfrac {a f - c d} {a e - ... | We have:
:$\dfrac {\d y} {\d x} = \map F {\dfrac {a x + b y + c} {d x + e y + f} }$
Make the substitutions:
:$x := z - h$
:$y := w - k$
We have:
:$\dfrac {\d x} {\d z} = 1$
:$\dfrac {\d y} {\d w} = 1$
Thus:
{{begin-eqn}}
{{eqn | l = \frac {\d w} {\d z}
| r = \map F {\frac {a \paren {z - h} + b \paren {w - k} ... | First Order ODE in form y' = F ((a x + b y + c) over (d x + e y + f)) | https://proofwiki.org/wiki/First_Order_ODE_in_form_y'_=_F_((a_x_+_b_y_+_c)_over_(d_x_+_e_y_+_f)) | https://proofwiki.org/wiki/First_Order_ODE_in_form_y'_=_F_((a_x_+_b_y_+_c)_over_(d_x_+_e_y_+_f)) | [
"First Order ODE in form y' = F ((a x + b y + c) over (d x + e y + f))",
"Examples of First Order ODEs"
] | [
"Definition:First Order Ordinary Differential Equation",
"Solution to Homogeneous Differential Equation"
] | [
"Definition:Homogeneous Differential Equation"
] |
proofwiki-10960 | First Order ODE in form y' = F ((a x + b y + c) over (d x + e y + f)) where a e = b d/Formulation 1 | The first order ODE:
:$(1): \quad \dfrac {\d y} {\d x} = \map F {\dfrac {a x + b y + c} {d x + e y + f} }$
such that:
:$a e = b d$
can be solved by substituting:
:$z = a x + b y$
to obtain:
:$\dfrac {\d z} {\d x} = b \map F {\dfrac {a z + a c} {d z + a f} } + a$
which can be solved by Solution to Separable Differential... | When $a e = b d$, it is not possible to make the substitutions:
:$x := z - h$
:$y := w - k$
where:
:$h = \dfrac {c e - b f} {a e - b d}$
:$k = \dfrac {a f - c d} {a e - b d}$
and so to use the technique of First Order ODE in form $y' = \map F {\dfrac {a x + b y + c} {d x + e y + f} }$.
So, we consider what needs to be ... | The [[Definition:First Order Ordinary Differential Equation|first order ODE]]:
:$(1): \quad \dfrac {\d y} {\d x} = \map F {\dfrac {a x + b y + c} {d x + e y + f} }$
such that:
:$a e = b d$
can be solved by substituting:
:$z = a x + b y$
to obtain:
:$\dfrac {\d z} {\d x} = b \map F {\dfrac {a z + a c} {d z + a f} }... | When $a e = b d$, it is not possible to make the substitutions:
:$x := z - h$
:$y := w - k$
where:
:$h = \dfrac {c e - b f} {a e - b d}$
:$k = \dfrac {a f - c d} {a e - b d}$
and so to use the technique of [[First Order ODE in form y' = F ((a x + b y + c) over (d x + e y + f))|First Order ODE in form $y' = \map F {\... | First Order ODE in form y' = F ((a x + b y + c) over (d x + e y + f)) where a e = b d/Formulation 1 | https://proofwiki.org/wiki/First_Order_ODE_in_form_y'_=_F_((a_x_+_b_y_+_c)_over_(d_x_+_e_y_+_f))_where_a_e_=_b_d/Formulation_1 | https://proofwiki.org/wiki/First_Order_ODE_in_form_y'_=_F_((a_x_+_b_y_+_c)_over_(d_x_+_e_y_+_f))_where_a_e_=_b_d/Formulation_1 | [
"First Order ODE in form y' = F ((a x + b y + c) over (d x + e y + f))"
] | [
"Definition:First Order Ordinary Differential Equation",
"Solution to Separable Differential Equation"
] | [
"First Order ODE in form y' = F ((a x + b y + c) over (d x + e y + f))",
"Definition:Separable Differential Equation",
"Definition:Separable Differential Equation",
"Definition:Separable Differential Equation"
] |
proofwiki-10961 | First Order ODE in form y' = F ((a x + b y + c) over (d x + e y + f)) where a e = b d/Formulation 2 | The first order ODE:
:$(1): \quad \dfrac {\d y} {\d x} = \map F {\dfrac {a x + b y + c} {d x + e y + f} }$
such that:
:$a e = b d$
can be solved by substituting:
:$z = d x + e y$
to obtain:
:$\dfrac {\d z} {\d x} = e \map F {\dfrac {b z + e c} {e z + e f} } + d$
which can be solved by Solution to Separable Differential... | When $a e = b d$, it is not possible to make the substitutions:
:$x := z - h$
:$y := w - k$
where:
:$h = \dfrac {c e - b f} {a e - b d}$
:$k = \dfrac {a f - c d} {a e - b d}$
and so to use the technique of First Order ODE in form $y' = \map F {\dfrac {a x + b y + c} {d x + e y + f} }$.
So, we consider what needs to be ... | The [[Definition:First Order Ordinary Differential Equation|first order ODE]]:
:$(1): \quad \dfrac {\d y} {\d x} = \map F {\dfrac {a x + b y + c} {d x + e y + f} }$
such that:
:$a e = b d$
can be solved by substituting:
:$z = d x + e y$
to obtain:
:$\dfrac {\d z} {\d x} = e \map F {\dfrac {b z + e c} {e z + e f} }... | When $a e = b d$, it is not possible to make the substitutions:
:$x := z - h$
:$y := w - k$
where:
:$h = \dfrac {c e - b f} {a e - b d}$
:$k = \dfrac {a f - c d} {a e - b d}$
and so to use the technique of [[First Order ODE in form y' = F ((a x + b y + c) over (d x + e y + f))|First Order ODE in form $y' = \map F {\... | First Order ODE in form y' = F ((a x + b y + c) over (d x + e y + f)) where a e = b d/Formulation 2 | https://proofwiki.org/wiki/First_Order_ODE_in_form_y'_=_F_((a_x_+_b_y_+_c)_over_(d_x_+_e_y_+_f))_where_a_e_=_b_d/Formulation_2 | https://proofwiki.org/wiki/First_Order_ODE_in_form_y'_=_F_((a_x_+_b_y_+_c)_over_(d_x_+_e_y_+_f))_where_a_e_=_b_d/Formulation_2 | [
"First Order ODE in form y' = F ((a x + b y + c) over (d x + e y + f))"
] | [
"Definition:First Order Ordinary Differential Equation",
"Solution to Separable Differential Equation"
] | [
"First Order ODE in form y' = F ((a x + b y + c) over (d x + e y + f))",
"Definition:Separable Differential Equation",
"Definition:Separable Differential Equation",
"Definition:Separable Differential Equation",
"Category:First Order ODE in form y' = F ((a x + b y + c) over (d x + e y + f))"
] |
proofwiki-10962 | First Order ODE/exp y dx + (x exp y + 2 y) dy = 0 | is an exact differential equation with solution:
:$x e^y + y^2 = C$ | Let:
:$\map M {x, y} = e^y$
:$\map N {x, y} = x e^y + 2 y$
Then:
{{begin-eqn}}
{{eqn | l = \dfrac {\partial M} {\partial y}
| r = e^y
| c =
}}
{{eqn | l = \dfrac {\partial N} {\partial x}
| r = e^y
| c =
}}
{{end-eqn}}
Thus $\dfrac {\partial M} {\partial y} = \dfrac {\partial N} {\partial x}$ ... | is an [[Definition:Exact Differential Equation|exact differential equation]] with [[Definition:General Solution to Differential Equation|solution]]:
:$x e^y + y^2 = C$ | Let:
:$\map M {x, y} = e^y$
:$\map N {x, y} = x e^y + 2 y$
Then:
{{begin-eqn}}
{{eqn | l = \dfrac {\partial M} {\partial y}
| r = e^y
| c =
}}
{{eqn | l = \dfrac {\partial N} {\partial x}
| r = e^y
| c =
}}
{{end-eqn}}
Thus $\dfrac {\partial M} {\partial y} = \dfrac {\partial N} {\partial x... | First Order ODE/exp y dx + (x exp y + 2 y) dy = 0 | https://proofwiki.org/wiki/First_Order_ODE/exp_y_dx_+_(x_exp_y_+_2_y)_dy_=_0 | https://proofwiki.org/wiki/First_Order_ODE/exp_y_dx_+_(x_exp_y_+_2_y)_dy_=_0 | [
"Examples of Exact Differential Equation",
"Examples of First Order ODEs"
] | [
"Definition:Exact Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Exact Differential Equation",
"Solution to Exact Differential Equation",
"Definition:Differential Equation/Solution/General Solution",
"Solution to Exact Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] |
proofwiki-10963 | First Order ODE/(x + (2 over y)) dy + y dx = 0 | is an exact differential equation with solution:
:$x y + 2 \ln y = C$ | Let:
:$\map M {x, y} = y$
:$\map N {x, y} = x + \dfrac 2 y$
Then:
{{begin-eqn}}
{{eqn | l = \dfrac {\partial M} {\partial y}
| r = 1
| c =
}}
{{eqn | l = \dfrac {\partial N} {\partial x}
| r = 1
| c =
}}
{{end-eqn}}
Thus $\dfrac {\partial M} {\partial y} = \dfrac {\partial N} {\partial x}$ and... | is an [[Definition:Exact Differential Equation|exact differential equation]] with [[Definition:General Solution to Differential Equation|solution]]:
:$x y + 2 \ln y = C$ | Let:
:$\map M {x, y} = y$
:$\map N {x, y} = x + \dfrac 2 y$
Then:
{{begin-eqn}}
{{eqn | l = \dfrac {\partial M} {\partial y}
| r = 1
| c =
}}
{{eqn | l = \dfrac {\partial N} {\partial x}
| r = 1
| c =
}}
{{end-eqn}}
Thus $\dfrac {\partial M} {\partial y} = \dfrac {\partial N} {\partial x}$ ... | First Order ODE/(x + (2 over y)) dy + y dx = 0 | https://proofwiki.org/wiki/First_Order_ODE/(x_+_(2_over_y))_dy_+_y_dx_=_0 | https://proofwiki.org/wiki/First_Order_ODE/(x_+_(2_over_y))_dy_+_y_dx_=_0 | [
"Examples of Exact Differential Equation",
"Examples of First Order ODEs"
] | [
"Definition:Exact Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Exact Differential Equation",
"Solution to Exact Differential Equation",
"Definition:Differential Equation/Solution/General Solution",
"Solution to Exact Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] |
proofwiki-10964 | First Order ODE/(y - x^3) dx + (x + y^3) dy = 0 | is an exact differential equation with solution:
:$4 x y - x^4 + y^4 = C$ | Let:
:$\map M {x, y} = y - x^3$
:$\map N {x, y} = x + y^3$
Then:
{{begin-eqn}}
{{eqn | l = \dfrac {\partial M} {\partial y}
| r = 1
| c =
}}
{{eqn | l = \dfrac {\partial N} {\partial x}
| r = 1
| c =
}}
{{end-eqn}}
Thus $\dfrac {\partial M} {\partial y} = \dfrac {\partial N} {\partial x}$ and ... | is an [[Definition:Exact Differential Equation|exact differential equation]] with [[Definition:General Solution to Differential Equation|solution]]:
:$4 x y - x^4 + y^4 = C$ | Let:
:$\map M {x, y} = y - x^3$
:$\map N {x, y} = x + y^3$
Then:
{{begin-eqn}}
{{eqn | l = \dfrac {\partial M} {\partial y}
| r = 1
| c =
}}
{{eqn | l = \dfrac {\partial N} {\partial x}
| r = 1
| c =
}}
{{end-eqn}}
Thus $\dfrac {\partial M} {\partial y} = \dfrac {\partial N} {\partial x}$ a... | First Order ODE/(y - x^3) dx + (x + y^3) dy = 0 | https://proofwiki.org/wiki/First_Order_ODE/(y_-_x^3)_dx_+_(x_+_y^3)_dy_=_0 | https://proofwiki.org/wiki/First_Order_ODE/(y_-_x^3)_dx_+_(x_+_y^3)_dy_=_0 | [
"Examples of Exact Differential Equation",
"Examples of First Order ODEs"
] | [
"Definition:Exact Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Exact Differential Equation",
"Solution to Exact Differential Equation",
"Definition:Differential Equation/Solution/General Solution",
"Solution to Exact Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] |
proofwiki-10965 | First Order ODE/(y + y cosine x y) dx + (x + x cosine x y) dy = 0 | is an exact differential equation with solution:
:$x y + \sin x y = C$ | Let:
:$\map M {x, y} = y + y \cos x y$
:$\map N {x, y} = x + x \cos x y$
Then:
{{begin-eqn}}
{{eqn | l = \dfrac {\partial M} {\partial y}
| r = 1 + \cos x y - x \sin x y
| c =
}}
{{eqn | l = \dfrac {\partial N} {\partial x}
| r = 1 + \cos x y - x \sin x y
| c =
}}
{{end-eqn}}
Thus $\dfrac {\... | is an [[Definition:Exact Differential Equation|exact differential equation]] with [[Definition:General Solution to Differential Equation|solution]]:
:$x y + \sin x y = C$ | Let:
:$\map M {x, y} = y + y \cos x y$
:$\map N {x, y} = x + x \cos x y$
Then:
{{begin-eqn}}
{{eqn | l = \dfrac {\partial M} {\partial y}
| r = 1 + \cos x y - x \sin x y
| c =
}}
{{eqn | l = \dfrac {\partial N} {\partial x}
| r = 1 + \cos x y - x \sin x y
| c =
}}
{{end-eqn}}
Thus $\dfrac... | First Order ODE/(y + y cosine x y) dx + (x + x cosine x y) dy = 0 | https://proofwiki.org/wiki/First_Order_ODE/(y_+_y_cosine_x_y)_dx_+_(x_+_x_cosine_x_y)_dy_=_0 | https://proofwiki.org/wiki/First_Order_ODE/(y_+_y_cosine_x_y)_dx_+_(x_+_x_cosine_x_y)_dy_=_0 | [
"Examples of Exact Differential Equation",
"Examples of First Order ODEs"
] | [
"Definition:Exact Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Exact Differential Equation",
"Solution to Exact Differential Equation",
"Definition:Differential Equation/Solution/General Solution",
"Solution to Exact Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] |
proofwiki-10966 | First Order ODE/(sine x sine y - x e^y) dy = (e^y + cosine x cosine y) dx | is an exact differential equation with solution:
:$\sin x \cos y + x e^y = C$ | First express $(1)$ in the form:
:$(2): \quad -\paren {e^y + \cos x \cos y} + \paren {\sin x \sin y - x e^y} \dfrac {\d y} {\d x}$
Let:
:$\map M {x, y} = -\paren {e^y + \cos x \cos y}$
:$\map N {x, y} = \sin x \sin y - x e^y$
Then:
{{begin-eqn}}
{{eqn | l = \dfrac {\partial M} {\partial y}
| r = -e^y + \cos x \si... | is an [[Definition:Exact Differential Equation|exact differential equation]] with [[Definition:General Solution to Differential Equation|solution]]:
:$\sin x \cos y + x e^y = C$ | First express $(1)$ in the form:
:$(2): \quad -\paren {e^y + \cos x \cos y} + \paren {\sin x \sin y - x e^y} \dfrac {\d y} {\d x}$
Let:
:$\map M {x, y} = -\paren {e^y + \cos x \cos y}$
:$\map N {x, y} = \sin x \sin y - x e^y$
Then:
{{begin-eqn}}
{{eqn | l = \dfrac {\partial M} {\partial y}
| r = -e^y + \cos x \... | First Order ODE/(sine x sine y - x e^y) dy = (e^y + cosine x cosine y) dx | https://proofwiki.org/wiki/First_Order_ODE/(sine_x_sine_y_-_x_e^y)_dy_=_(e^y_+_cosine_x_cosine_y)_dx | https://proofwiki.org/wiki/First_Order_ODE/(sine_x_sine_y_-_x_e^y)_dy_=_(e^y_+_cosine_x_cosine_y)_dx | [
"Examples of Exact Differential Equation",
"Examples of First Order ODEs"
] | [
"Definition:Exact Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Exact Differential Equation",
"Solution to Exact Differential Equation",
"Definition:Differential Equation/Solution/General Solution",
"Solution to Exact Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] |
proofwiki-10967 | First Order ODE/-1 over y sine x over y dx + x over y^2 sine x over y dy | is an exact differential equation with solution:
:$\dfrac x y = C$ | Let:
:$\map M {x, y} = -\dfrac 1 y \sin \dfrac x y$
:$\map N {x, y} = \dfrac x {y^2} \sin \dfrac x y$
Then:
{{begin-eqn}}
{{eqn | l = \frac {\partial M} {\partial y}
| r = \frac 1 {y^2} \sin \frac x y + \paren {-\frac 1 y} \paren {-\frac 1 {y^2} } x \cos \frac x y
| c =
}}
{{eqn | r = \frac 1 {y^2} \sin \f... | is an [[Definition:Exact Differential Equation|exact differential equation]] with [[Definition:General Solution to Differential Equation|solution]]:
:$\dfrac x y = C$ | Let:
:$\map M {x, y} = -\dfrac 1 y \sin \dfrac x y$
:$\map N {x, y} = \dfrac x {y^2} \sin \dfrac x y$
Then:
{{begin-eqn}}
{{eqn | l = \frac {\partial M} {\partial y}
| r = \frac 1 {y^2} \sin \frac x y + \paren {-\frac 1 y} \paren {-\frac 1 {y^2} } x \cos \frac x y
| c =
}}
{{eqn | r = \frac 1 {y^2} \sin \... | First Order ODE/-1 over y sine x over y dx + x over y^2 sine x over y dy | https://proofwiki.org/wiki/First_Order_ODE/-1_over_y_sine_x_over_y_dx_+_x_over_y^2_sine_x_over_y_dy | https://proofwiki.org/wiki/First_Order_ODE/-1_over_y_sine_x_over_y_dx_+_x_over_y^2_sine_x_over_y_dy | [
"Examples of Exact Differential Equation",
"Examples of First Order ODEs"
] | [
"Definition:Exact Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Exact Differential Equation",
"Solution to Exact Differential Equation",
"Definition:Differential Equation/Solution/General Solution",
"Integration by Substitution",
"Solution to Exact Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] |
proofwiki-10968 | First Order ODE/(2 x y^3 + y cosine x) dx + (3 x^2 y^2 + sine x) dy | is an exact differential equation with solution:
:$x^2 y^3 + y \sin x = C$ | Let:
:$\map M {x, y} = 2 x y^3 + y \cos x$
:$\map N {x, y} = 3 x^2 y^2 + \sin x$
Then:
{{begin-eqn}}
{{eqn | l = \frac {\partial M} {\partial y}
| r = 2 x \cdot 3 y^2 + \cos x
| c =
}}
{{eqn | r = 6 x y^2 + \cos x
| c =
}}
{{eqn | l = \frac {\partial N} {\partial x}
| r = 3 x^2 \cdot 2 y^2 + \... | is an [[Definition:Exact Differential Equation|exact differential equation]] with [[Definition:General Solution to Differential Equation|solution]]:
:$x^2 y^3 + y \sin x = C$ | Let:
:$\map M {x, y} = 2 x y^3 + y \cos x$
:$\map N {x, y} = 3 x^2 y^2 + \sin x$
Then:
{{begin-eqn}}
{{eqn | l = \frac {\partial M} {\partial y}
| r = 2 x \cdot 3 y^2 + \cos x
| c =
}}
{{eqn | r = 6 x y^2 + \cos x
| c =
}}
{{eqn | l = \frac {\partial N} {\partial x}
| r = 3 x^2 \cdot 2 y^2 + ... | First Order ODE/(2 x y^3 + y cosine x) dx + (3 x^2 y^2 + sine x) dy | https://proofwiki.org/wiki/First_Order_ODE/(2_x_y^3_+_y_cosine_x)_dx_+_(3_x^2_y^2_+_sine_x)_dy | https://proofwiki.org/wiki/First_Order_ODE/(2_x_y^3_+_y_cosine_x)_dx_+_(3_x^2_y^2_+_sine_x)_dy | [
"Examples of Exact Differential Equation",
"Examples of First Order ODEs"
] | [
"Definition:Exact Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Exact Differential Equation",
"Solution to Exact Differential Equation",
"Definition:Differential Equation/Solution/General Solution",
"Solution to Exact Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] |
proofwiki-10969 | First Order ODE/dx = (y over (1 - x^2 y^2)) dx + (x over (1 - x^2 y^2)) dy | is an exact differential equation with solution:
:$\map \ln {\dfrac {1 + x y} {1 - x y} } - 2 x = C$ | First express $(1)$ in the form:
:$(2): \quad \paren {\dfrac y {1 - x^2 y^2} - 1} + \paren {\dfrac x {1 - x^2 y^2} } \dfrac {\d y} {\d x}$
Let:
:$\map M {x, y} = \dfrac y {1 - x^2 y^2} - 1$
:$\map N {x, y} = \dfrac x {1 - x^2 y^2}$
Then:
{{begin-eqn}}
{{eqn | l = \frac {\partial M} {\partial y}
| r = \frac {\pare... | is an [[Definition:Exact Differential Equation|exact differential equation]] with [[Definition:General Solution to Differential Equation|solution]]:
:$\map \ln {\dfrac {1 + x y} {1 - x y} } - 2 x = C$ | First express $(1)$ in the form:
:$(2): \quad \paren {\dfrac y {1 - x^2 y^2} - 1} + \paren {\dfrac x {1 - x^2 y^2} } \dfrac {\d y} {\d x}$
Let:
:$\map M {x, y} = \dfrac y {1 - x^2 y^2} - 1$
:$\map N {x, y} = \dfrac x {1 - x^2 y^2}$
Then:
{{begin-eqn}}
{{eqn | l = \frac {\partial M} {\partial y}
| r = \frac {\p... | First Order ODE/dx = (y over (1 - x^2 y^2)) dx + (x over (1 - x^2 y^2)) dy | https://proofwiki.org/wiki/First_Order_ODE/dx_=_(y_over_(1_-_x^2_y^2))_dx_+_(x_over_(1_-_x^2_y^2))_dy | https://proofwiki.org/wiki/First_Order_ODE/dx_=_(y_over_(1_-_x^2_y^2))_dx_+_(x_over_(1_-_x^2_y^2))_dy | [
"Examples of Exact Differential Equation",
"Examples of First Order ODEs"
] | [
"Definition:Exact Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Quotient Rule for Derivatives",
"Definition:Exact Differential Equation",
"Solution to Exact Differential Equation",
"Definition:Differential Equation/Solution/General Solution",
"Integral Operator is Linear",
"Integration by Partial Fractions",
"Integral Operator is Linear",
"Difference of Logarithm... |
proofwiki-10970 | Limit Point of Subset is Limit Point of Set | Let $T = \struct {S, \tau}$ be a topological space.
Let $A, B$ be subset of $S$ such that
:$A \subseteq B$
Let $x$ be a point of $S$.
Then:
:if $x$ is limit point of $A$, then $x$ is limit point of $B$. | Assume $x$ is limit point of $A$.
By definition of limit point it suffices to prove
:$\forall U \in \tau: x \in U \implies B \cap \paren {U \setminus \set x} \ne \O$
Let $U \in \tau$ such that
:$x \in U$
By definition of limit point:
:$A \cap \paren {U \setminus \set x} \ne \O$
By Set Intersection Preserves Subsets/Cor... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A, B$ be [[Definition:Subset|subset]] of $S$ such that
:$A \subseteq B$
Let $x$ be a [[Definition:Element|point]] of $S$.
Then:
:if $x$ is [[Definition:Limit Point of Set|limit point]] of $A$, then $x$ is [[Definition:Limit Po... | Assume $x$ is [[Definition:Limit Point of Set|limit point]] of $A$.
By definition of [[Definition:Limit Point of Set|limit point]] it suffices to prove
:$\forall U \in \tau: x \in U \implies B \cap \paren {U \setminus \set x} \ne \O$
Let $U \in \tau$ such that
:$x \in U$
By definition of [[Definition:Limit Point of ... | Limit Point of Subset is Limit Point of Set | https://proofwiki.org/wiki/Limit_Point_of_Subset_is_Limit_Point_of_Set | https://proofwiki.org/wiki/Limit_Point_of_Subset_is_Limit_Point_of_Set | [
"Limit Points"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Element",
"Definition:Limit Point/Topology/Set",
"Definition:Limit Point/Topology/Set"
] | [
"Definition:Limit Point/Topology/Set",
"Definition:Limit Point/Topology/Set",
"Definition:Limit Point/Topology/Set",
"Set Intersection Preserves Subsets/Corollary",
"Category:Limit Points"
] |
proofwiki-10971 | Existence of Infinitely Many Integrating Factors | Let the first order ordinary differential equation:
:$(1): \quad \map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$
be such that $M$ and $N$ are real functions of two variables which are ''not'' homogeneous functions of the same degree.
Suppose $(1)$ has an integrating factor.
Then $(1)$ has an infinite number of ... | Let $\map F f$ be any function of $f$ which is an integrating factor of $(1)$.
Then:
:$\ds \mu \map F f \paren {\map M {x, y} \rd x + \map N {x, y} \rd y} = \map F f \rd f = \map \d {\int \map F f \rd f}$
so $\mu \map F f$ is also an integrating factor.
{{qed}} | Let the [[Definition:First Order Ordinary Differential Equation|first order ordinary differential equation]]:
:$(1): \quad \map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$
be such that $M$ and $N$ are [[Definition:Real Function|real functions]] of two variables which are ''not'' [[Definition:Homogeneous Real Fun... | Let $\map F f$ be any function of $f$ which is an [[Definition:Integrating Factor|integrating factor]] of $(1)$.
Then:
:$\ds \mu \map F f \paren {\map M {x, y} \rd x + \map N {x, y} \rd y} = \map F f \rd f = \map \d {\int \map F f \rd f}$
so $\mu \map F f$ is also an [[Definition:Integrating Factor|integrating factor]... | Existence of Infinitely Many Integrating Factors | https://proofwiki.org/wiki/Existence_of_Infinitely_Many_Integrating_Factors | https://proofwiki.org/wiki/Existence_of_Infinitely_Many_Integrating_Factors | [
"Integrating Factors"
] | [
"Definition:First Order Ordinary Differential Equation",
"Definition:Real Function",
"Definition:Homogeneous Function/Real Space",
"Definition:Homogeneous Function/Real Space/Degree",
"Definition:Integrating Factor",
"Definition:Infinite",
"Definition:Integrating Factor"
] | [
"Definition:Integrating Factor",
"Definition:Integrating Factor"
] |
proofwiki-10972 | Integrating Factor for First Order ODE/Preliminary Work | Let the first order ordinary differential equation:
:$(1): \quad \map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$
be non-homogeneous and not exact.
Let $\map \mu x$ be an integrating factor for $(1)$.
Let:
:$\map P {x, y} := \dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x}$
Then:
:$\dfrac 1 \m... | Let us for ease of manipulation express $(1)$ in the form of differentials:
:$(2): \quad \map M {x, y} \rd x + \map N {x, y} \rd y = 0$
Let $\mu$ be an integrating factor for $(2)$.
Then, by definition:
:$\mu \, \map M {x, y} \rd x + \mu \, \map N {x, y} \rd y = 0$
is an exact differential equation.
By Solution to Exac... | Let the [[Definition:First Order Ordinary Differential Equation|first order ordinary differential equation]]:
:$(1): \quad \map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$
be non-[[Definition:Homogeneous Differential Equation|homogeneous]] and not [[Definition:Exact Differential Equation|exact]].
Let $\map \mu ... | Let us for ease of manipulation express $(1)$ in the form of [[Definition:Differential of Real Function|differentials]]:
:$(2): \quad \map M {x, y} \rd x + \map N {x, y} \rd y = 0$
Let $\mu$ be an [[Definition:Integrating Factor|integrating factor]] for $(2)$.
Then, by definition:
:$\mu \, \map M {x, y} \rd x + \mu \... | Integrating Factor for First Order ODE/Preliminary Work | https://proofwiki.org/wiki/Integrating_Factor_for_First_Order_ODE/Preliminary_Work | https://proofwiki.org/wiki/Integrating_Factor_for_First_Order_ODE/Preliminary_Work | [
"Integrating Factors"
] | [
"Definition:First Order Ordinary Differential Equation",
"Definition:Homogeneous Differential Equation",
"Definition:Exact Differential Equation",
"Definition:Integrating Factor"
] | [
"Definition:Differential of Mapping/Real Function",
"Definition:Integrating Factor",
"Definition:Exact Differential Equation",
"Solution to Exact Differential Equation",
"Product Rule for Derivatives"
] |
proofwiki-10973 | Integrating Factor for First Order ODE/Function of One Variable | Suppose that:
:$\map g x = \dfrac {\dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x} } {\map N {x, y} }$
is a function of $x$ only.
Then:
:$\map \mu x = e^{\int \map g x \rd x}$
is an integrating factor for $(1)$.
Similarly, suppose that:
:$\map h y = \dfrac {\dfrac {\partial M} {\partial y} - \dfrac ... | === Preliminary Work ===
{{:Integrating Factor for First Order ODE/Preliminary Work}} | Suppose that:
:$\map g x = \dfrac {\dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x} } {\map N {x, y} }$
is a [[Definition:Real Function|function]] of $x$ only.
Then:
:$\map \mu x = e^{\int \map g x \rd x}$
is an [[Definition:Integrating Factor|integrating factor]] for $(1)$.
Similarly, suppose tha... | === [[Integrating Factor for First Order ODE/Preliminary Work|Preliminary Work]] ===
{{:Integrating Factor for First Order ODE/Preliminary Work}} | Integrating Factor for First Order ODE/Function of One Variable | https://proofwiki.org/wiki/Integrating_Factor_for_First_Order_ODE/Function_of_One_Variable | https://proofwiki.org/wiki/Integrating_Factor_for_First_Order_ODE/Function_of_One_Variable | [
"Integrating Factors"
] | [
"Definition:Real Function",
"Definition:Integrating Factor",
"Definition:Real Function",
"Definition:Integrating Factor"
] | [
"Integrating Factor for First Order ODE/Preliminary Work"
] |
proofwiki-10974 | Integrating Factor for First Order ODE/Function of Sum of Variables | Suppose that:
:$\map g z = \dfrac {\dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x} } {\map N {x, y} - \map M {x, y} }$
is a function of $z$, where $z = x + y$.
Then:
:$\map \mu {x + y} = \map \mu z = e^{\int \map g z \rd z}$
is an integrating factor for $(1)$. | === Preliminary Work ===
{{:Integrating Factor for First Order ODE/Preliminary Work}} | Suppose that:
:$\map g z = \dfrac {\dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x} } {\map N {x, y} - \map M {x, y} }$
is a [[Definition:Real Function|function]] of $z$, where $z = x + y$.
Then:
:$\map \mu {x + y} = \map \mu z = e^{\int \map g z \rd z}$
is an [[Definition:Integrating Factor|integra... | === [[Integrating Factor for First Order ODE/Preliminary Work|Preliminary Work]] ===
{{:Integrating Factor for First Order ODE/Preliminary Work}} | Integrating Factor for First Order ODE/Function of Sum of Variables | https://proofwiki.org/wiki/Integrating_Factor_for_First_Order_ODE/Function_of_Sum_of_Variables | https://proofwiki.org/wiki/Integrating_Factor_for_First_Order_ODE/Function_of_Sum_of_Variables | [
"Integrating Factors"
] | [
"Definition:Real Function",
"Definition:Integrating Factor"
] | [
"Integrating Factor for First Order ODE/Preliminary Work"
] |
proofwiki-10975 | Integrating Factor for First Order ODE/Function of Product of Variables | :$\map g z = \dfrac {\dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x}} {N y - M x}$
is a function of $z$, where $z = x y$.
Then:
:$\map \mu {x y} = \map \mu z = e^{\int \map g z \d z}$
is an integrating factor for $(1)$. | === Preliminary Work ===
{{:Integrating Factor for First Order ODE/Preliminary Work}} | :$\map g z = \dfrac {\dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x}} {N y - M x}$
is a [[Definition:Real Function|function]] of $z$, where $z = x y$.
Then:
:$\map \mu {x y} = \map \mu z = e^{\int \map g z \d z}$
is an [[Definition:Integrating Factor|integrating factor]] for $(1)$. | === [[Integrating Factor for First Order ODE/Preliminary Work|Preliminary Work]] ===
{{:Integrating Factor for First Order ODE/Preliminary Work}} | Integrating Factor for First Order ODE/Function of Product of Variables | https://proofwiki.org/wiki/Integrating_Factor_for_First_Order_ODE/Function_of_Product_of_Variables | https://proofwiki.org/wiki/Integrating_Factor_for_First_Order_ODE/Function_of_Product_of_Variables | [
"Integrating Factors"
] | [
"Definition:Real Function",
"Definition:Integrating Factor"
] | [
"Integrating Factor for First Order ODE/Preliminary Work"
] |
proofwiki-10976 | Set of Condensation Points of Union is Union of Sets of Condensation Points | Let $T = \left({S, \tau}\right)$ be a topological space.
Let $A, B$ be subsets of $S$.
Then:
:$\left({A \cup B}\right)^0 = A^0 \cup B^0$ | === Set of Condensation Points of Union Subset Union of Sets of Condensation Points ===
Let $x \in \left({A \cup B}\right)^0$.
By definition of set of condensation points:
:$x$ is condensation point of $A \cup B$
By Lemma:
:$x$ is condensation point of $A$ or $x$ is condensation point of $B$
By definition of set of con... | Let $T = \left({S, \tau}\right)$ be a [[Definition:Topological Space|topological space]].
Let $A, B$ be [[Definition:Subset|subsets]] of $S$.
Then:
:$\left({A \cup B}\right)^0 = A^0 \cup B^0$ | === Set of Condensation Points of Union Subset Union of Sets of Condensation Points ===
Let $x \in \left({A \cup B}\right)^0$.
By definition of [[Definition:Set of Condensation Points|set of condensation points]]:
:$x$ is [[Definition:Condensation Point|condensation point]] of $A \cup B$
By [[Set of Condensation Poi... | Set of Condensation Points of Union is Union of Sets of Condensation Points | https://proofwiki.org/wiki/Set_of_Condensation_Points_of_Union_is_Union_of_Sets_of_Condensation_Points | https://proofwiki.org/wiki/Set_of_Condensation_Points_of_Union_is_Union_of_Sets_of_Condensation_Points | [
"Condensation Points"
] | [
"Definition:Topological Space",
"Definition:Subset"
] | [
"Definition:Set of Condensation Points",
"Definition:Condensation Point",
"Set of Condensation Points of Union is Union of Sets of Condensation Points/Lemma",
"Definition:Condensation Point",
"Definition:Condensation Point",
"Definition:Set of Condensation Points",
"Definition:Set Union"
] |
proofwiki-10977 | Set of Condensation Points of Union is Union of Sets of Condensation Points/Lemma | Let $x$ be a point of $S$.
Then:
:if $x$ is condensation point of $A \cup B$,
:then $x$ is condensation point of $A$ or $x$ is condensation point of $B$. | Assume $x$ is condensation point of $A \cup B$.
{{AimForCont}}
:$x$ is not condensation point of $A$ and $x$ is not condensation point of $B$.
By definition of condensation point:
:$\exists U_1 \in \tau: x \in U_1 \land A \cap U_1$ is countable
By definition of condensation point:
:$\exists U_2 \in \tau: x \in U_2 \lan... | Let $x$ be a [[Definition:Element|point]] of $S$.
Then:
:if $x$ is [[Definition:Condensation Point|condensation point]] of $A \cup B$,
:then $x$ is [[Definition:Condensation Point|condensation point]] of $A$ or $x$ is [[Definition:Condensation Point|condensation point]] of $B$. | Assume $x$ is [[Definition:Condensation Point|condensation point]] of $A \cup B$.
{{AimForCont}}
:$x$ is not [[Definition:Condensation Point|condensation point]] of $A$ and $x$ is not [[Definition:Condensation Point|condensation point]] of $B$.
By definition of [[Definition:Condensation Point|condensation point]]:
:$... | Set of Condensation Points of Union is Union of Sets of Condensation Points/Lemma | https://proofwiki.org/wiki/Set_of_Condensation_Points_of_Union_is_Union_of_Sets_of_Condensation_Points/Lemma | https://proofwiki.org/wiki/Set_of_Condensation_Points_of_Union_is_Union_of_Sets_of_Condensation_Points/Lemma | [
"Condensation Points"
] | [
"Definition:Element",
"Definition:Condensation Point",
"Definition:Condensation Point",
"Definition:Condensation Point"
] | [
"Definition:Condensation Point",
"Definition:Condensation Point",
"Definition:Condensation Point",
"Definition:Condensation Point",
"Definition:Countable Set",
"Definition:Condensation Point",
"Definition:Countable Set",
"Definition:Set Intersection",
"Definition:Topological Space",
"Intersection ... |
proofwiki-10978 | Integrating Factor for First Order ODE/Conclusion | Let the first order ordinary differential equation:
:$(1): \quad \map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$
be non-homogeneous and not exact.
Let $\map \mu {x, y}$be an integrating factor for $(1)$.
If one of these is the case:
:$\mu$ is a function of $x$ only
:$\mu$ is a function of $y$ only
:$\mu$ is a f... | We have an equation of the form:
:$\dfrac 1 \mu \dfrac {\d \mu} {\d w} = \map f w$
which is what you get when you apply the Chain Rule for Derivatives and Derivative of Logarithm Function to:
:$\dfrac {\map \d {\ln \mu} } {\d w} = \map f w$
Thus:
:$\ds \ln \mu = \int \map f w \rd w$
and so:
:$\mu = e^{\int \map f w \rd... | Let the [[Definition:First Order Ordinary Differential Equation|first order ordinary differential equation]]:
:$(1): \quad \map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$
be non-[[Definition:Homogeneous Differential Equation|homogeneous]] and not [[Definition:Exact Differential Equation|exact]].
Let $\map \mu ... | We have an equation of the form:
:$\dfrac 1 \mu \dfrac {\d \mu} {\d w} = \map f w$
which is what you get when you apply the [[Chain Rule for Derivatives]] and [[Derivative of Logarithm Function]] to:
:$\dfrac {\map \d {\ln \mu} } {\d w} = \map f w$
Thus:
:$\ds \ln \mu = \int \map f w \rd w$
and so:
:$\mu = e^{\int \m... | Integrating Factor for First Order ODE/Conclusion | https://proofwiki.org/wiki/Integrating_Factor_for_First_Order_ODE/Conclusion | https://proofwiki.org/wiki/Integrating_Factor_for_First_Order_ODE/Conclusion | [
"Integrating Factors"
] | [
"Definition:First Order Ordinary Differential Equation",
"Definition:Homogeneous Differential Equation",
"Definition:Exact Differential Equation",
"Definition:Integrating Factor"
] | [
"Derivative of Composite Function",
"Derivative of Logarithm Function"
] |
proofwiki-10979 | First Order ODE/y dx + (x^2 y - x) dy = 0 | The first order ODE:
:$(1): \quad y \rd x + \paren {x^2 y - x} \rd y = 0$
has the general solution:
:$\dfrac {y^2} 2 - \dfrac y x = C$ | We note that $(1)$ is in the form:
:$\map M {x, y} \d x + \map N {x, y} \d y = 0$
but that $(1)$ is not exact.
So, let:
:$\map M {x, y} = y$
:$\map N {x, y} = x^2 y - x$
Let:
:$\map P {x, y} = \dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x}$
Thus:
{{begin-eqn}}
{{eqn | l = \map P {x, y}
| r = ... | The [[Definition:First Order ODE|first order ODE]]:
:$(1): \quad y \rd x + \paren {x^2 y - x} \rd y = 0$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$\dfrac {y^2} 2 - \dfrac y x = C$ | We note that $(1)$ is in the form:
:$\map M {x, y} \d x + \map N {x, y} \d y = 0$
but that $(1)$ is not [[Definition:Exact Differential Equation|exact]].
So, let:
:$\map M {x, y} = y$
:$\map N {x, y} = x^2 y - x$
Let:
:$\map P {x, y} = \dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x}$
Thus:
{{beg... | First Order ODE/y dx + (x^2 y - x) dy = 0/Proof 1 | https://proofwiki.org/wiki/First_Order_ODE/y_dx_+_(x^2_y_-_x)_dy_=_0 | https://proofwiki.org/wiki/First_Order_ODE/y_dx_+_(x^2_y_-_x)_dy_=_0/Proof_1 | [
"Examples of First Order ODEs",
"First Order ODE/y dx + (x^2 y - x) dy = 0"
] | [
"Definition:First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Exact Differential Equation",
"Integrating Factor for First Order ODE/Function of One Variable",
"Definition:Integrating Factor",
"Definition:Exact Differential Equation",
"First Order ODE/(y over x^2) dx + (y - 1 over x) dy = 0"
] |
proofwiki-10980 | First Order ODE/y dx + (x^2 y - x) dy = 0 | The first order ODE:
:$(1): \quad y \rd x + \paren {x^2 y - x} \rd y = 0$
has the general solution:
:$\dfrac {y^2} 2 - \dfrac y x = C$ | Rearranging, we have:
:$x^2 y \rd y - \paren {x \rd y - y \rd x} = 0$
Aiming to use Quotient Rule for Differentials, divide by $x^2$:
:$y \rd y = \dfrac {x \rd y - y \rd x} {x^2}$
So from Quotient Rule for Differentials: Formulation 1
:$y \rd y = \map \d {\dfrac y x}$
from which the solution immediately drops:
:$\dfrac... | The [[Definition:First Order ODE|first order ODE]]:
:$(1): \quad y \rd x + \paren {x^2 y - x} \rd y = 0$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$\dfrac {y^2} 2 - \dfrac y x = C$ | Rearranging, we have:
:$x^2 y \rd y - \paren {x \rd y - y \rd x} = 0$
Aiming to use [[Quotient Rule for Differentials]], divide by $x^2$:
:$y \rd y = \dfrac {x \rd y - y \rd x} {x^2}$
So from [[Quotient Rule for Differentials/Formulation 1|Quotient Rule for Differentials: Formulation 1]]
:$y \rd y = \map \d {\dfrac y... | First Order ODE/y dx + (x^2 y - x) dy = 0/Proof 2 | https://proofwiki.org/wiki/First_Order_ODE/y_dx_+_(x^2_y_-_x)_dy_=_0 | https://proofwiki.org/wiki/First_Order_ODE/y_dx_+_(x^2_y_-_x)_dy_=_0/Proof_2 | [
"Examples of First Order ODEs",
"First Order ODE/y dx + (x^2 y - x) dy = 0"
] | [
"Definition:First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Quotient Rule for Differentials",
"Quotient Rule for Differentials/Formulation 1"
] |
proofwiki-10981 | Set of Condensation Points of Countable Set is Empty | Let $T = \struct {S, \tau}$ be a topological space.
Let $A$ be a subset of $S$.
Then:
:if $A$ is countable,
:then $A^0 = \O$. | Assume
:$A$ is countable.
{{AimForCont}}
:$A^0 \ne \O$
By definition of empty set:
:$\exists x: x \in A^0$
Then by definition of set of condensation points:
:$x$ is a condensation point of $A$.
This contradicts Lemma.
Thus the result follows by Proof by Contradiction.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A$ be a [[Definition:Subset|subset]] of $S$.
Then:
:if $A$ is [[Definition:Countable Set|countable]],
:then $A^0 = \O$. | Assume
:$A$ is [[Definition:Countable Set|countable]].
{{AimForCont}}
:$A^0 \ne \O$
By definition of [[Definition:Empty Set|empty set]]:
:$\exists x: x \in A^0$
Then by definition of [[Definition:Set of Condensation Points|set of condensation points]]:
:$x$ is a [[Definition:Condensation Point|condensation point]] o... | Set of Condensation Points of Countable Set is Empty | https://proofwiki.org/wiki/Set_of_Condensation_Points_of_Countable_Set_is_Empty | https://proofwiki.org/wiki/Set_of_Condensation_Points_of_Countable_Set_is_Empty | [
"Condensation Points"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Countable Set"
] | [
"Definition:Countable Set",
"Definition:Empty Set",
"Definition:Set of Condensation Points",
"Definition:Condensation Point",
"Set of Condensation Points of Countable Set is Empty/Lemma",
"Proof by Contradiction"
] |
proofwiki-10982 | Set of Condensation Points of Countable Set is Empty/Lemma | :if $A$ is countable,
:then there exists no point $x$ of $S$ such that $x$ is a condensation point of $A$. | Assume
:$A$ is countable.
{{AimForCont}} there exists a point $x$ of $S$ such that
: $x$ is a condensation point of $A$
By definition of topological space:
:$S \in \tau$
Then by definition of condensation point:
:$A \cap S$ is uncountable
By Intersection with Subset is Subset:
:$A \cap S = A$
$A$ is countable contradic... | :if $A$ is [[Definition:Countable Set|countable]],
:then there exists no [[Definition:Element|point]] $x$ of $S$ such that $x$ is a [[Definition:Condensation Point|condensation point]] of $A$. | Assume
:$A$ is [[Definition:Countable Set|countable]].
{{AimForCont}} there exists a [[Definition:Element|point]] $x$ of $S$ such that
: $x$ is a [[Definition:Condensation Point|condensation point]] of $A$
By definition of [[Definition:Topological Space|topological space]]:
:$S \in \tau$
Then by definition of [[Defi... | Set of Condensation Points of Countable Set is Empty/Lemma | https://proofwiki.org/wiki/Set_of_Condensation_Points_of_Countable_Set_is_Empty/Lemma | https://proofwiki.org/wiki/Set_of_Condensation_Points_of_Countable_Set_is_Empty/Lemma | [
"Condensation Points"
] | [
"Definition:Countable Set",
"Definition:Element",
"Definition:Condensation Point"
] | [
"Definition:Countable Set",
"Definition:Element",
"Definition:Condensation Point",
"Definition:Topological Space",
"Definition:Condensation Point",
"Definition:Countable Set",
"Intersection with Subset is Subset",
"Definition:Countable Set",
"Definition:Countable Set",
"Proof by Contradiction"
] |
proofwiki-10983 | First Order ODE/(3 x^2 - y^2) dy - 2 x y dx = 0 | The first order ODE:
:$(1): \quad \paren {3 x^2 - y^2} \rd y - 2 x y \rd x = 0$
has the general solution:
:$\dfrac 1 y - \dfrac {x^2} {y^3} = C$ | We note that $(1)$ is in the form:
:$\map M {x, y} \rd x + \map N {x, y} \rd y = 0$
but that $(1)$ is not exact.
So, let:
:$\map M {x, y} = -2 x y$
:$\map N {x, y} = 3 x^2 - y^2$
Let:
:$\map P {x, y} = \dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x}$
Thus:
{{begin-eqn}}
{{eqn | l = \map P {x, y}
... | The [[Definition:First Order ODE|first order ODE]]:
:$(1): \quad \paren {3 x^2 - y^2} \rd y - 2 x y \rd x = 0$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$\dfrac 1 y - \dfrac {x^2} {y^3} = C$ | We note that $(1)$ is in the form:
:$\map M {x, y} \rd x + \map N {x, y} \rd y = 0$
but that $(1)$ is not [[Definition:Exact Differential Equation|exact]].
So, let:
:$\map M {x, y} = -2 x y$
:$\map N {x, y} = 3 x^2 - y^2$
Let:
:$\map P {x, y} = \dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x}$
Th... | First Order ODE/(3 x^2 - y^2) dy - 2 x y dx = 0 | https://proofwiki.org/wiki/First_Order_ODE/(3_x^2_-_y^2)_dy_-_2_x_y_dx_=_0 | https://proofwiki.org/wiki/First_Order_ODE/(3_x^2_-_y^2)_dy_-_2_x_y_dx_=_0 | [
"Examples of First Order ODEs"
] | [
"Definition:First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Exact Differential Equation",
"Integrating Factor for First Order ODE/Function of One Variable",
"Definition:Integrating Factor",
"Definition:Exact Differential Equation",
"First Order ODE/(3 x^2 over y^4 - 1 over y^2) dy - 2 x over y^3 dx = 0"
] |
proofwiki-10984 | First Order ODE/(x y - 1) dx + (x^2 - x y) dy = 0 | The first order ODE:
:$(1): \quad \paren {x y - 1} \rd x + \paren {x^2 - x y} \rd y = 0$
has the general solution:
:$x y - \ln x - \dfrac {y^2} 2 + C$ | We note that $(1)$ is in the form:
:$\map M {x, y} \rd x + \map N {x, y} \rd y = 0$
but that $(1)$ is not exact.
So, let:
:$\map M {x, y} = x y - 1$
:$\map N {x, y} = x^2 - x y = x \paren {x - y}$
Let:
:$\map P {x, y} = \dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x}$
Thus:
{{begin-eqn}}
{{eqn | l =... | The [[Definition:First Order ODE|first order ODE]]:
:$(1): \quad \paren {x y - 1} \rd x + \paren {x^2 - x y} \rd y = 0$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$x y - \ln x - \dfrac {y^2} 2 + C$ | We note that $(1)$ is in the form:
:$\map M {x, y} \rd x + \map N {x, y} \rd y = 0$
but that $(1)$ is not [[Definition:Exact Differential Equation|exact]].
So, let:
:$\map M {x, y} = x y - 1$
:$\map N {x, y} = x^2 - x y = x \paren {x - y}$
Let:
:$\map P {x, y} = \dfrac {\partial M} {\partial y} - \dfrac {\partial N}... | First Order ODE/(x y - 1) dx + (x^2 - x y) dy = 0 | https://proofwiki.org/wiki/First_Order_ODE/(x_y_-_1)_dx_+_(x^2_-_x_y)_dy_=_0 | https://proofwiki.org/wiki/First_Order_ODE/(x_y_-_1)_dx_+_(x^2_-_x_y)_dy_=_0 | [
"Examples of First Order ODEs"
] | [
"Definition:First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Exact Differential Equation",
"Integrating Factor for First Order ODE/Function of One Variable",
"Definition:Integrating Factor",
"Definition:Exact Differential Equation",
"First Order ODE/(y - 1 over x) dx + (x - y) dy = 0"
] |
proofwiki-10985 | First Order ODE/y dx + x dy + 3 x^3 y^4 dy | The first order ODE:
:$(1): \quad y \rd x + x \rd y + 3 x^3 y^4 \rd y = 0$
has the general solution:
:$-\dfrac 1 {2 x^2 y^2} + \dfrac {3 y^2} 2 = C$ | We note that $(1)$ is in the form:
:$\map M {x, y} \rd x + \map N {x, y} \rd y = 0$
but that $(1)$ is not exact.
So, let:
:$\map M {x, y} = y$
:$\map N {x, y} = 3 x^3 y^4 + x = x \paren {3 x^2 y^4 + 1}$
Let:
:$\map P {x, y} = \dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x}$
Thus:
{{begin-eqn}}
{{eqn... | The [[Definition:First Order ODE|first order ODE]]:
:$(1): \quad y \rd x + x \rd y + 3 x^3 y^4 \rd y = 0$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$-\dfrac 1 {2 x^2 y^2} + \dfrac {3 y^2} 2 = C$ | We note that $(1)$ is in the form:
:$\map M {x, y} \rd x + \map N {x, y} \rd y = 0$
but that $(1)$ is not [[Definition:Exact Differential Equation|exact]].
So, let:
:$\map M {x, y} = y$
:$\map N {x, y} = 3 x^3 y^4 + x = x \paren {3 x^2 y^4 + 1}$
Let:
:$\map P {x, y} = \dfrac {\partial M} {\partial y} - \dfrac {\part... | First Order ODE/y dx + x dy + 3 x^3 y^4 dy | https://proofwiki.org/wiki/First_Order_ODE/y_dx_+_x_dy_+_3_x^3_y^4_dy | https://proofwiki.org/wiki/First_Order_ODE/y_dx_+_x_dy_+_3_x^3_y^4_dy | [
"First Order ODE/y dx + x dy + 3 x^3 y^4 dy",
"Examples of First Order ODEs"
] | [
"Definition:First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Exact Differential Equation",
"Integrating Factor for First Order ODE/Function of Product of Variables",
"Definition:Integrating Factor",
"Definition:Exact Differential Equation",
"First Order ODE/1 over x^3 y^2 dx + (1 over x^2 y^3 + 3 y) dy = 0"
] |
proofwiki-10986 | Curved Mirror producing Parallel Rays is Paraboloid | Let $M$ be a curved mirror embedded in a real cartesian $3$- space.
Let there be a source of light at the origin.
Let $M$ reflect the light in a beam parallel to the $x$-axis.
Then $M$ is the solid of revolution produced by rotating about the $x$-axis the parabola whose equation is:
:$y^2 = 2 c x + c^2$ | The mirror will have the shape of a surface of revolution generated by revolving a curve $APB$ in the cartesian plane around the $x$-axis.
Let $P = \tuple {x, y}$ be an arbitrary point on $APB$.
:500px
From the Law of Reflection:
:$\alpha = \beta$
From Parallelism implies Equal Corresponding Angles:
:$\phi = \beta$
Fro... | Let $M$ be a [[Definition:Curve|curved]] [[Definition:Mirror|mirror]] embedded in a [[Definition:Real Cartesian Space|real cartesian $3$- space]].
Let there be a source of [[Definition:Light (Radiation)|light]] at the [[Definition:Origin|origin]].
Let $M$ reflect the [[Definition:Light (Radiation)|light]] in a beam [... | The [[Definition:Mirror|mirror]] will have the shape of a [[Definition:Surface of Revolution|surface of revolution]] generated by revolving a [[Definition:Curve|curve]] $APB$ in the [[Definition:Cartesian Plane|cartesian plane]] around the [[Definition:X-Axis|$x$-axis]].
Let $P = \tuple {x, y}$ be an arbitrary [[Defin... | Curved Mirror producing Parallel Rays is Paraboloid | https://proofwiki.org/wiki/Curved_Mirror_producing_Parallel_Rays_is_Paraboloid | https://proofwiki.org/wiki/Curved_Mirror_producing_Parallel_Rays_is_Paraboloid | [
"Optics"
] | [
"Definition:Line/Curve",
"Definition:Mirror",
"Definition:Cartesian Product/Cartesian Space/Real Cartesian Space",
"Definition:Light (Radiation)",
"Definition:Coordinate System/Origin",
"Definition:Light (Radiation)",
"Definition:Parallel (Geometry)/Lines",
"Definition:Axis/X-Axis",
"Definition:Soli... | [
"Definition:Mirror",
"Definition:Surface of Revolution",
"Definition:Line/Curve",
"Definition:Cartesian Plane",
"Definition:Axis/X-Axis",
"Definition:Point",
"File:ParabolicMirror.png",
"Law of Reflection",
"Parallelism implies Equal Corresponding Angles",
"External Angle of Triangle equals Sum of... |
proofwiki-10987 | First Order ODE/x dy - y dx = (1 + y^2) dy | The first order ODE:
:$(1): \quad x \rd y - y \rd x = \paren {1 + y^2} \rd y$
has the general solution:
:$\dfrac x y = \dfrac 1 y - y + C$ | Rearranging, we have:
:$\dfrac {y \rd x - x \rd y} {y^2} = -\paren {\dfrac 1 {y^2} + 1} \rd y$
From the Quotient Rule for Derivatives:
:$\map \d {\dfrac x y} = \dfrac {y \rd x - x \rd y} {y^2}$
from which:
:$\map \d {\dfrac x y} = -\paren {\dfrac 1 {y^2} + 1} \rd y$
Hence the result:
:$\dfrac x y = \dfrac 1 y - y + C$
... | The [[Definition:First Order ODE|first order ODE]]:
:$(1): \quad x \rd y - y \rd x = \paren {1 + y^2} \rd y$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$\dfrac x y = \dfrac 1 y - y + C$ | Rearranging, we have:
:$\dfrac {y \rd x - x \rd y} {y^2} = -\paren {\dfrac 1 {y^2} + 1} \rd y$
From the [[Quotient Rule for Derivatives]]:
:$\map \d {\dfrac x y} = \dfrac {y \rd x - x \rd y} {y^2}$
from which:
:$\map \d {\dfrac x y} = -\paren {\dfrac 1 {y^2} + 1} \rd y$
Hence the result:
:$\dfrac x y = \dfrac 1 y - ... | First Order ODE/x dy - y dx = (1 + y^2) dy | https://proofwiki.org/wiki/First_Order_ODE/x_dy_-_y_dx_=_(1_+_y^2)_dy | https://proofwiki.org/wiki/First_Order_ODE/x_dy_-_y_dx_=_(1_+_y^2)_dy | [
"Examples of First Order ODEs"
] | [
"Definition:First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Quotient Rule for Derivatives"
] |
proofwiki-10988 | Differential of Sum of Squares | :$\map \d {x^2 + y^2} = 2 \paren {x \map \rd x + y \map \rd y}$ | {{begin-eqn}}
{{eqn | l = \map \rd {x^2 + y^2; \vec h}
| r = 2 x \vec h_1 + 2 y \vec h_2
| c = {{Defof|Differential of Mapping|subdef = Real-Valued Function}}
}}
{{eqn | r = 2 x \map \rd {x; \vec h} + 2 y \map \rd {y; \vec h}
}}
{{eqn | r = 2 \paren {x \map \rd {x; \vec h} + y \map \rd {y; \vec h} }
}}
{{en... | :$\map \d {x^2 + y^2} = 2 \paren {x \map \rd x + y \map \rd y}$ | {{begin-eqn}}
{{eqn | l = \map \rd {x^2 + y^2; \vec h}
| r = 2 x \vec h_1 + 2 y \vec h_2
| c = {{Defof|Differential of Mapping|subdef = Real-Valued Function}}
}}
{{eqn | r = 2 x \map \rd {x; \vec h} + 2 y \map \rd {y; \vec h}
}}
{{eqn | r = 2 \paren {x \map \rd {x; \vec h} + y \map \rd {y; \vec h} }
}}
{{en... | Differential of Sum of Squares | https://proofwiki.org/wiki/Differential_of_Sum_of_Squares | https://proofwiki.org/wiki/Differential_of_Sum_of_Squares | [
"Differentials"
] | [] | [] |
proofwiki-10989 | First Order ODE/y dx + (x^2 y - x) dy = 0/Proof 1 | {{:First Order ODE/y dx + (x^2 y - x) dy = 0}}
This can also be presented in the form:
:$\dfrac {\d y} {\d x} + \dfrac y {x^2 y - x}$ | We note that $(1)$ is in the form:
:$\map M {x, y} \d x + \map N {x, y} \d y = 0$
but that $(1)$ is not exact.
So, let:
:$\map M {x, y} = y$
:$\map N {x, y} = x^2 y - x$
Let:
:$\map P {x, y} = \dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x}$
Thus:
{{begin-eqn}}
{{eqn | l = \map P {x, y}
| r = ... | {{:First Order ODE/y dx + (x^2 y - x) dy = 0}}
This can also be presented in the form:
:$\dfrac {\d y} {\d x} + \dfrac y {x^2 y - x}$ | We note that $(1)$ is in the form:
:$\map M {x, y} \d x + \map N {x, y} \d y = 0$
but that $(1)$ is not [[Definition:Exact Differential Equation|exact]].
So, let:
:$\map M {x, y} = y$
:$\map N {x, y} = x^2 y - x$
Let:
:$\map P {x, y} = \dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x}$
Thus:
{{beg... | First Order ODE/y dx + (x^2 y - x) dy = 0/Proof 1 | https://proofwiki.org/wiki/First_Order_ODE/y_dx_+_(x^2_y_-_x)_dy_=_0/Proof_1 | https://proofwiki.org/wiki/First_Order_ODE/y_dx_+_(x^2_y_-_x)_dy_=_0/Proof_1 | [
"First Order ODE/y dx + (x^2 y - x) dy = 0"
] | [] | [
"Definition:Exact Differential Equation",
"Integrating Factor for First Order ODE/Function of One Variable",
"Definition:Integrating Factor",
"Definition:Exact Differential Equation",
"First Order ODE/(y over x^2) dx + (y - 1 over x) dy = 0"
] |
proofwiki-10990 | First Order ODE/y dx - x dy = x y^3 dy | The first order ODE:
:$(1): \quad y \rd x - x \rd y = x y^3 \rd y$
has the general solution:
:$\ln \dfrac x y = \dfrac {y^3} 3 + C$ | Rearranging, we have:
:$\dfrac {y \rd x - x \rd y} {x y} = y^2 \rd y$
From Differential of Logarithm of Quotient:
:$\map \d {\ln \dfrac y x} = \dfrac {y \rd x - x \rd y} {x y}$
from which:
:$\map \d {\ln \dfrac x y} = y^2 \rd y$
Hence the result:
:$\ln \dfrac x y = \dfrac {y^3} 3 + C$
{{qed}} | The [[Definition:First Order ODE|first order ODE]]:
:$(1): \quad y \rd x - x \rd y = x y^3 \rd y$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$\ln \dfrac x y = \dfrac {y^3} 3 + C$ | Rearranging, we have:
:$\dfrac {y \rd x - x \rd y} {x y} = y^2 \rd y$
From [[Differential of Logarithm of Quotient]]:
:$\map \d {\ln \dfrac y x} = \dfrac {y \rd x - x \rd y} {x y}$
from which:
:$\map \d {\ln \dfrac x y} = y^2 \rd y$
Hence the result:
:$\ln \dfrac x y = \dfrac {y^3} 3 + C$
{{qed}} | First Order ODE/y dx - x dy = x y^3 dy | https://proofwiki.org/wiki/First_Order_ODE/y_dx_-_x_dy_=_x_y^3_dy | https://proofwiki.org/wiki/First_Order_ODE/y_dx_-_x_dy_=_x_y^3_dy | [
"Examples of First Order ODEs"
] | [
"Definition:First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Differential of Logarithm of Quotient"
] |
proofwiki-10991 | First Order ODE/x dy = (x^5 + x^3 y^2 + y) dx | The first order ODE:
:$(1): \quad x \rd y = \paren {x^5 + x^3 y^2 + y} \rd x$
has the general solution:
:$\arctan \dfrac x y = -\dfrac {x^4} 4 + C$ | Rearranging, we have:
:$y \rd x - x \rd y = -\paren {x^2 + y^2} x^3 \rd x$
from which:
:$\dfrac {y \rd x - x \rd y} {x^2 + y^2} = - x^3 \rd x$
From Differential of Arctangent of Quotient:
:$\map \d {\arctan \dfrac x y} = \dfrac {y \rd x - x \rd y} {x^2 + y^2}$
from which $(1)$ evolves into:
:$\map \d {\arctan \dfrac x ... | The [[Definition:First Order ODE|first order ODE]]:
:$(1): \quad x \rd y = \paren {x^5 + x^3 y^2 + y} \rd x$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$\arctan \dfrac x y = -\dfrac {x^4} 4 + C$ | Rearranging, we have:
:$y \rd x - x \rd y = -\paren {x^2 + y^2} x^3 \rd x$
from which:
:$\dfrac {y \rd x - x \rd y} {x^2 + y^2} = - x^3 \rd x$
From [[Differential of Arctangent of Quotient]]:
:$\map \d {\arctan \dfrac x y} = \dfrac {y \rd x - x \rd y} {x^2 + y^2}$
from which $(1)$ evolves into:
:$\map \d {\arctan \d... | First Order ODE/x dy = (x^5 + x^3 y^2 + y) dx | https://proofwiki.org/wiki/First_Order_ODE/x_dy_=_(x^5_+_x^3_y^2_+_y)_dx | https://proofwiki.org/wiki/First_Order_ODE/x_dy_=_(x^5_+_x^3_y^2_+_y)_dx | [
"Examples of First Order ODEs"
] | [
"Definition:First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Differential of Arctangent of Quotient"
] |
proofwiki-10992 | First Order ODE/(x + y) dx = (x - y) dy/Proof 1 | The first order ordinary differential equation:
:$(1): \quad \paren {x + y} \rd x = \paren {x - y} \rd y$
is a homogeneous differential equation with solution:
:$\arctan \dfrac y x = \ln \sqrt{x^2 + y^2} + C$ | Let:
:$\map M {x, y} = x + y$
:$\map N {x, y} = x - y$
We have that:
:$\map M {t x, t y} = t x + t y = t \paren {x + y} = t \map M {x, y}$
:$\map N {t x, t y} = t x - t y = t \paren {x - y} = t \map N {x, y}$
Thus both $M$ and $N$ are homogeneous functions of degree $1$.
Thus by definition $(1)$ is a homogeneous di... | The [[Definition:First Order Ordinary Differential Equation|first order ordinary differential equation]]:
:$(1): \quad \paren {x + y} \rd x = \paren {x - y} \rd y$
is a [[Definition:Homogeneous Differential Equation|homogeneous differential equation]] with [[Definition:General Solution to Differential Equation|solut... | Let:
:$\map M {x, y} = x + y$
:$\map N {x, y} = x - y$
We have that:
:$\map M {t x, t y} = t x + t y = t \paren {x + y} = t \map M {x, y}$
:$\map N {t x, t y} = t x - t y = t \paren {x - y} = t \map N {x, y}$
Thus both $M$ and $N$ are [[Definition:Homogeneous Real Function|homogeneous functions]] of [[Definition... | First Order ODE/(x + y) dx = (x - y) dy/Proof 1 | https://proofwiki.org/wiki/First_Order_ODE/(x_+_y)_dx_=_(x_-_y)_dy/Proof_1 | https://proofwiki.org/wiki/First_Order_ODE/(x_+_y)_dx_=_(x_-_y)_dy/Proof_1 | [
"Examples of First Order ODEs"
] | [
"Definition:First Order Ordinary Differential Equation",
"Definition:Homogeneous Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Function/Real Space",
"Definition:Homogeneous Function/Real Space/Degree",
"Definition:Homogeneous Differential Equation",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Solution to Homogeneous Differential Equation",
"Primitive of Reciprocal of x squared pl... |
proofwiki-10993 | First Order ODE/x dy = (y + x^2 + 9 y^2) dx | The first order ODE:
:$(1): \quad x \rd y = \paren {y + x^2 + 9 y^2} \rd x$
has the general solution:
:$\map \arctan {\dfrac {3 y} x} = 3 x + C$ | Divide both sides of $(1)$ by $x^2 \rd x$ to get:
:$\dfrac 1 x \dfrac {\d y} {\d x} = \dfrac 1 x \paren {\dfrac y x } + 1 + 9 \paren {\dfrac y x}^2$
Now apply the substitution:
:$y = u x$
This implies then that:
:$\dfrac {\d y} {\d x} = u + x \dfrac {\d u} {\d x}$
Now substitute everything into $(1)$ to get:
{{begin-eq... | The [[Definition:First Order ODE|first order ODE]]:
:$(1): \quad x \rd y = \paren {y + x^2 + 9 y^2} \rd x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$\map \arctan {\dfrac {3 y} x} = 3 x + C$ | Divide both sides of $(1)$ by $x^2 \rd x$ to get:
:$\dfrac 1 x \dfrac {\d y} {\d x} = \dfrac 1 x \paren {\dfrac y x } + 1 + 9 \paren {\dfrac y x}^2$
Now apply the substitution:
:$y = u x$
This implies then that:
:$\dfrac {\d y} {\d x} = u + x \dfrac {\d u} {\d x}$
Now substitute everything into $(1)$ to get:
{{begi... | First Order ODE/x dy = (y + x^2 + 9 y^2) dx/Proof 1 | https://proofwiki.org/wiki/First_Order_ODE/x_dy_=_(y_+_x^2_+_9_y^2)_dx | https://proofwiki.org/wiki/First_Order_ODE/x_dy_=_(y_+_x^2_+_9_y^2)_dx/Proof_1 | [
"Examples of First Order ODEs",
"First Order ODE/x dy = (y + x^2 + 9 y^2) dx"
] | [
"Definition:First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Solution to Separable Differential Equation"
] |
proofwiki-10994 | First Order ODE/x dy = (y + x^2 + 9 y^2) dx | The first order ODE:
:$(1): \quad x \rd y = \paren {y + x^2 + 9 y^2} \rd x$
has the general solution:
:$\map \arctan {\dfrac {3 y} x} = 3 x + C$ | Let $z = \map \arctan {3y / x}$.
Then:
:$\dfrac {\partial z} {\partial x} = \dfrac 1 {1 + \paren {3 y / x}^2} \dfrac {-3 y} {x^2} = \dfrac {-3 y} {x^2 + 9 y^2}$
:$\dfrac {\partial z} {\partial y} = \dfrac 1 {1 + \paren {3 y / x}^2} \dfrac 3 x = \dfrac 3 {x^2 + 9 y^2}$
So:
:$\d z = \dfrac {3 x \rd y - 3 y \rd x} {x^2 + ... | The [[Definition:First Order ODE|first order ODE]]:
:$(1): \quad x \rd y = \paren {y + x^2 + 9 y^2} \rd x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$\map \arctan {\dfrac {3 y} x} = 3 x + C$ | Let $z = \map \arctan {3y / x}$.
Then:
:$\dfrac {\partial z} {\partial x} = \dfrac 1 {1 + \paren {3 y / x}^2} \dfrac {-3 y} {x^2} = \dfrac {-3 y} {x^2 + 9 y^2}$
:$\dfrac {\partial z} {\partial y} = \dfrac 1 {1 + \paren {3 y / x}^2} \dfrac 3 x = \dfrac 3 {x^2 + 9 y^2}$
So:
:$\d z = \dfrac {3 x \rd y - 3 y \rd x} {x^2... | First Order ODE/x dy = (y + x^2 + 9 y^2) dx/Proof 2 | https://proofwiki.org/wiki/First_Order_ODE/x_dy_=_(y_+_x^2_+_9_y^2)_dx | https://proofwiki.org/wiki/First_Order_ODE/x_dy_=_(y_+_x^2_+_9_y^2)_dx/Proof_2 | [
"Examples of First Order ODEs",
"First Order ODE/x dy = (y + x^2 + 9 y^2) dx"
] | [
"Definition:First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Differential of Arctangent of Quotient"
] |
proofwiki-10995 | Linear First Order ODE/y' + (y over x) = 3 x | The linear first order ODE:
:$\dfrac {\d y} {\d x} + \dfrac y x = 3 x$
has the general solution:
:$x y = x^3 + C$
or:
:$y = x^2 + \dfrac C x$ | This is a special case of:
:Linear First Order ODE: $\dfrac {\d y} {\d x} + \dfrac y x = k x^n$
where $k = 3$ and $n = 1$, yielding:
:$y = x^2 + \dfrac C x$
Multiplying through by $x$ reveals:
:$x y = x^3 + C$
{{qed}} | The [[Definition:Linear First Order ODE|linear first order ODE]]:
:$\dfrac {\d y} {\d x} + \dfrac y x = 3 x$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$x y = x^3 + C$
or:
:$y = x^2 + \dfrac C x$ | This is a special case of:
:[[Linear First Order ODE/y' + (y over x) = k x^n|Linear First Order ODE: $\dfrac {\d y} {\d x} + \dfrac y x = k x^n$]]
where $k = 3$ and $n = 1$, yielding:
:$y = x^2 + \dfrac C x$
Multiplying through by $x$ reveals:
:$x y = x^3 + C$
{{qed}} | Linear First Order ODE/y' + (y over x) = 3 x | https://proofwiki.org/wiki/Linear_First_Order_ODE/y'_+_(y_over_x)_=_3_x | https://proofwiki.org/wiki/Linear_First_Order_ODE/y'_+_(y_over_x)_=_3_x | [
"Examples of Linear First Order ODEs"
] | [
"Definition:Linear First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Linear First Order ODE/y' + (y over x) = k x^n"
] |
proofwiki-10996 | Linear First Order ODE/x y' - 3 y = x^4 | The linear first order ODE:
:$(1): \quad x \dfrac {\d y} {\d x} - 3y = x^4$
has the general solution:
:$y = x^4 + \dfrac C {x^3}$ | Rearranging $(1)$:
:$(2): \quad \dfrac {\d y} {\d x} + \paren {-\dfrac 3 x} y = x^3$
$(2)$ is a linear first order ODE in the form:
:$\dfrac {\d y} {\d x} + \map P x y = \map Q x$
where:
:$\map P x = -\dfrac 3 x$
:$\map Q x = x^3$
Thus:
{{begin-eqn}}
{{eqn | l = \int \map P x \rd x
| r = \int -\frac 3 x \rd x
... | The [[Definition:Linear First Order ODE|linear first order ODE]]:
:$(1): \quad x \dfrac {\d y} {\d x} - 3y = x^4$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$y = x^4 + \dfrac C {x^3}$ | Rearranging $(1)$:
:$(2): \quad \dfrac {\d y} {\d x} + \paren {-\dfrac 3 x} y = x^3$
$(2)$ 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:
:$\map P x = -\dfrac 3 x$
:$\map Q x = x^3$
Thus:
{{begin-eqn}}
{{eqn | l = \int \map P x \... | Linear First Order ODE/x y' - 3 y = x^4 | https://proofwiki.org/wiki/Linear_First_Order_ODE/x_y'_-_3_y_=_x^4 | https://proofwiki.org/wiki/Linear_First_Order_ODE/x_y'_-_3_y_=_x^4 | [
"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",
"Definition:Differential Equation/Solution/General Solution"
] |
proofwiki-10997 | Linear First Order ODE/y' + y = 1 over (1 + exp 2 x) | The linear first order ODE:
:$(1): \quad y' + y = \dfrac 1 {1 + e^{2 x} }$
has the general solution:
:$y = e^{-x} \map \arctan {e^x} + C e^{-x}$ | $(1)$ is in the form:
:$\dfrac {\d y} {\d x} + \map P x y = \map Q x$
where $\map P x = 1$.
Thus:
{{begin-eqn}}
{{eqn | l = \int \map P x \rd x
| r = \int \rd x
| c =
}}
{{eqn | r = x
| c =
}}
{{eqn | ll= \leadsto
| l = e^{\int P \rd x}
| r = e^x
| c =
}}
{{end-eqn}}
Thus from Sol... | The [[Definition:Linear First Order ODE|linear first order ODE]]:
:$(1): \quad y' + y = \dfrac 1 {1 + e^{2 x} }$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$y = e^{-x} \map \arctan {e^x} + C e^{-x}$ | $(1)$ is in the form:
:$\dfrac {\d y} {\d x} + \map P x y = \map Q x$
where $\map P x = 1$.
Thus:
{{begin-eqn}}
{{eqn | l = \int \map P x \rd x
| r = \int \rd x
| c =
}}
{{eqn | r = x
| c =
}}
{{eqn | ll= \leadsto
| l = e^{\int P \rd x}
| r = e^x
| c =
}}
{{end-eqn}}
Thus from [... | Linear First Order ODE/y' + y = 1 over (1 + exp 2 x) | https://proofwiki.org/wiki/Linear_First_Order_ODE/y'_+_y_=_1_over_(1_+_exp_2_x) | https://proofwiki.org/wiki/Linear_First_Order_ODE/y'_+_y_=_1_over_(1_+_exp_2_x) | [
"Examples of Linear First Order ODEs"
] | [
"Definition:Linear First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Solution to Linear First Order Ordinary Differential Equation/Solution by Integrating Factor",
"Definition:Differential Equation/Solution/General Solution",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form"
] |
proofwiki-10998 | Linear First Order ODE/(1 + x^2) dy + 2 x y dx = cotangent x dx | The linear first order ODE:
:$(1): \quad \paren {1 + x^2} \rd y + 2 x y \rd x = \cot x \rd x$
has the general solution:
:$y = \dfrac {\map \ln {\sin x} } {1 + x^2} + \dfrac C {1 + x^2}$ | $(1)$ can be written as:
:$(2): \quad \paren {1 + x^2} \dfrac {\rd y} {\rd x} + 2 x y = \cot x$
We notice straight away that:
:$\dfrac {\rd} {\rd x} \paren {1 + x^2} = 2 x$
and so:
:$\dfrac {\rd} {\rd x} \paren {1 + x^2} y = \cot x$
Thus the general solution becomes:
{{begin-eqn}}
{{eqn | l = \paren {1 + x^2} y
|... | The [[Definition:Linear First Order ODE|linear first order ODE]]:
:$(1): \quad \paren {1 + x^2} \rd y + 2 x y \rd x = \cot x \rd x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = \dfrac {\map \ln {\sin x} } {1 + x^2} + \dfrac C {1 + x^2}$ | $(1)$ can be written as:
:$(2): \quad \paren {1 + x^2} \dfrac {\rd y} {\rd x} + 2 x y = \cot x$
We notice straight away that:
:$\dfrac {\rd} {\rd x} \paren {1 + x^2} = 2 x$
and so:
:$\dfrac {\rd} {\rd x} \paren {1 + x^2} y = \cot x$
Thus the [[Definition:General Solution to Differential Equation|general solution]] ... | Linear First Order ODE/(1 + x^2) dy + 2 x y dx = cotangent x dx | https://proofwiki.org/wiki/Linear_First_Order_ODE/(1_+_x^2)_dy_+_2_x_y_dx_=_cotangent_x_dx | https://proofwiki.org/wiki/Linear_First_Order_ODE/(1_+_x^2)_dy_+_2_x_y_dx_=_cotangent_x_dx | [
"Examples of Linear First Order ODEs"
] | [
"Definition:Linear First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Differential Equation/Solution/General Solution"
] |
proofwiki-10999 | Linear First Order ODE/y' + y = 2 x exp -x + x^2 | The linear first order ODE:
:$(1): \quad y' + y = 2 x e^{-x} + x^2$
has the general solution:
:$y = x^2 e^{-x} + x^2 - 2 x + 2 + C e^{-x}$ | $(1)$ is in the form:
:$\dfrac {\d y} {\d x} + \map P x y = \map Q x$
where:
:$\map P x = 1$
Thus:
{{begin-eqn}}
{{eqn | l = \int \map P x \rd x
| r = \int 1 \rd x
| c =
}}
{{eqn | r = x
| c =
}}
{{eqn | ll= \leadsto
| l = e^{\int P \rd x}
| r = e^x
| c =
}}
{{end-eqn}}
Thus from ... | The [[Definition:Linear First Order ODE|linear first order ODE]]:
:$(1): \quad y' + y = 2 x e^{-x} + x^2$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$y = x^2 e^{-x} + x^2 - 2 x + 2 + C e^{-x}$ | $(1)$ is in the form:
:$\dfrac {\d y} {\d x} + \map P x y = \map Q x$
where:
:$\map P x = 1$
Thus:
{{begin-eqn}}
{{eqn | l = \int \map P x \rd x
| r = \int 1 \rd x
| c =
}}
{{eqn | r = x
| c =
}}
{{eqn | ll= \leadsto
| l = e^{\int P \rd x}
| r = e^x
| c =
}}
{{end-eqn}}
Thus fro... | Linear First Order ODE/y' + y = 2 x exp -x + x^2 | https://proofwiki.org/wiki/Linear_First_Order_ODE/y'_+_y_=_2_x_exp_-x_+_x^2 | https://proofwiki.org/wiki/Linear_First_Order_ODE/y'_+_y_=_2_x_exp_-x_+_x^2 | [
"Examples of Linear First Order ODEs"
] | [
"Definition:Linear First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Solution to Linear First Order Ordinary Differential Equation/Solution by Integrating Factor",
"Definition:Differential Equation/Solution/General Solution"
] |
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