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-16600 | Partial Sums of P-adic Expansion forms Coherent Sequence | Let $p$ be a prime number.
Let $\ds \sum_{n \mathop = 0}^\infty d_n p^n$ be a $p$-adic expansion.
Let $\sequence {\alpha_n}$ be the sequence of partial sums; that is:
:$\forall n \in \N :\alpha_n = \ds \sum_{i \mathop = 0}^n d_i p^i$.
Then $\sequence {\alpha_n}$ is a coherent sequence. | From the definition of a coherent sequence, it needs to be shown that $\sequence {\alpha_n}$ is a sequence of integers such that:
:$(1): \quad \forall n \in \N: 0 \le \alpha_n < p^{n + 1}$
:$(2): \quad \forall n \in \N: \alpha_{n + 1} \equiv \alpha_n \pmod {p^{n + 1}}$
That the sequence $\sequence {\alpha_n}$ is a sequ... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\ds \sum_{n \mathop = 0}^\infty d_n p^n$ be a [[Definition:P-adic Expansion|$p$-adic expansion]].
Let $\sequence {\alpha_n}$ be the [[Definition:Sequence of Partial Sums|sequence of partial sums]]; that is:
:$\forall n \in \N :\alpha_n = \ds \sum_{i \mathop ... | From the definition of a [[Definition:P-adically Coherent Sequence|coherent sequence]], it needs to be shown that $\sequence {\alpha_n}$ is a [[Definition:Sequence|sequence]] of [[Definition:Integer|integers]] such that:
:$(1): \quad \forall n \in \N: 0 \le \alpha_n < p^{n + 1}$
:$(2): \quad \forall n \in \N: \alpha_{n... | Partial Sums of P-adic Expansion forms Coherent Sequence | https://proofwiki.org/wiki/Partial_Sums_of_P-adic_Expansion_forms_Coherent_Sequence | https://proofwiki.org/wiki/Partial_Sums_of_P-adic_Expansion_forms_Coherent_Sequence | [
"P-adic Number Theory"
] | [
"Definition:Prime Number",
"Definition:P-adic Expansion",
"Definition:Series/Sequence of Partial Sums",
"Definition:P-adically Coherent Sequence"
] | [
"Definition:P-adically Coherent Sequence",
"Definition:Sequence",
"Definition:Integer",
"Definition:Sequence",
"Definition:Sequence",
"Definition:Integer",
"Definition:Series",
"Definition:Term",
"Definition:Summation",
"Definition:Integer"
] |
proofwiki-16601 | P-adic Expansion is a Cauchy Sequence in P-adic Norm/Represents a P-adic Number | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic Numbers.
Then the sequence of partial sums of the series:
:$\ds \sum_{n \mathop = m}^\infty d_n p^n$
represents a $p$-adic number of $\struct {\Q_p,\norm {\,\cdot\,}_p}$. | From P-adic Expansion is a Cauchy Sequence in P-adic Norm, the sequence of partial sums of the series:
:$\ds \sum_{n \mathop = m}^\infty d_n p^n$
is a Cauchy Sequence.
Then the sequence of partial sums of the series:
:$\ds \sum_{n \mathop = m}^\infty d_n p^n$
is a representative of a $p$-adic number by definition.
{{qe... | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic Numbers]].
Then the [[Definition:Sequence of Partial Sums|sequence of partial sums]] of the [[Definition:Series|series]]:
:$\ds \sum_{n \mathop = m}^\infty d_n p^n$
[[Definition:Representative of P-adic Number|repres... | From [[P-adic Expansion is a Cauchy Sequence in P-adic Norm]], the [[Definition:Sequence of Partial Sums|sequence of partial sums]] of the [[Definition:Series|series]]:
:$\ds \sum_{n \mathop = m}^\infty d_n p^n$
is a [[Definition:Cauchy Sequence (Normed Division Ring)|Cauchy Sequence]].
Then the [[Definition:Sequence ... | P-adic Expansion is a Cauchy Sequence in P-adic Norm/Represents a P-adic Number | https://proofwiki.org/wiki/P-adic_Expansion_is_a_Cauchy_Sequence_in_P-adic_Norm/Represents_a_P-adic_Number | https://proofwiki.org/wiki/P-adic_Expansion_is_a_Cauchy_Sequence_in_P-adic_Norm/Represents_a_P-adic_Number | [
"P-adic Expansion is a Cauchy Sequence in P-adic Norm"
] | [
"Definition:Valued Field of P-adic Numbers",
"Definition:Series/Sequence of Partial Sums",
"Definition:Series",
"Definition:P-adic Number/Representative",
"Definition:P-adic Number"
] | [
"P-adic Expansion is a Cauchy Sequence in P-adic Norm",
"Definition:Series/Sequence of Partial Sums",
"Definition:Series",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Series/Sequence of Partial Sums",
"Definition:Series",
"Definition:P-adic Number/Representative",
"Definition:P-adic... |
proofwiki-16602 | Gibbs Phenomenon | The Fourier series overshoots at a jump discontinuity, and adding more terms to the sum does not cause this overshoot to die out.
:500px
:500px
{{finish|Anyone care to flesh this out? I've got bored with it.}} | {{ProofWanted|Working on it}}
{{Namedfor|Josiah Willard Gibbs|cat = Gibbs}} | The [[Definition:Fourier Series|Fourier series]] overshoots at a [[Definition:Jump Discontinuity|jump discontinuity]], and adding more terms to the sum does not cause this overshoot to die out.
:[[File:Gibbs-Phenomenon-9.png|500px]]
:[[File:Gibbs-Phenomenon-19.png|500px]]
{{finish|Anyone care to flesh this out? I've... | {{ProofWanted|Working on it}}
{{Namedfor|Josiah Willard Gibbs|cat = Gibbs}} | Gibbs Phenomenon | https://proofwiki.org/wiki/Gibbs_Phenomenon | https://proofwiki.org/wiki/Gibbs_Phenomenon | [
"Fourier Series"
] | [
"Definition:Fourier Series",
"Definition:Discontinuity (Real Analysis)/Jump",
"File:Gibbs-Phenomenon-9.png",
"File:Gibbs-Phenomenon-19.png"
] | [] |
proofwiki-16603 | Root of Equation e^x (x - 1) = e^-x (x + 1) | The equation:
:$e^x \paren {x - 1} = e^{-x} \paren {x + 1}$
has a root:
:$x = 1 \cdotp 19966 \, 78640 \, 25773 \, 4 \ldots$ | Let $\map f x = e^x \paren {x - 1} - e^{-x} \paren {x + 1}$.
Then if $\map f c = 0$, $c$ is a root of $e^x \paren {x - 1} = e^{-x} \paren {x + 1}$.
Notice that:
:$\map f 1 = e^1 \paren {1 - 1} - e^{-1} \paren {1 + 1} = -\dfrac 2 e < 0$
:$\map f 2 = e^2 \paren {2 - 1} - e^{-2} \paren {2 + 1} = e^2 - \dfrac 3 {e^2} > 0$
... | The equation:
:$e^x \paren {x - 1} = e^{-x} \paren {x + 1}$
has a root:
:$x = 1 \cdotp 19966 \, 78640 \, 25773 \, 4 \ldots$ | Let $\map f x = e^x \paren {x - 1} - e^{-x} \paren {x + 1}$.
Then if $\map f c = 0$, $c$ is a root of $e^x \paren {x - 1} = e^{-x} \paren {x + 1}$.
Notice that:
:$\map f 1 = e^1 \paren {1 - 1} - e^{-1} \paren {1 + 1} = -\dfrac 2 e < 0$
:$\map f 2 = e^2 \paren {2 - 1} - e^{-2} \paren {2 + 1} = e^2 - \dfrac 3 {e^2} ... | Root of Equation e^x (x - 1) = e^-x (x + 1) | https://proofwiki.org/wiki/Root_of_Equation_e^x_(x_-_1)_=_e^-x_(x_+_1) | https://proofwiki.org/wiki/Root_of_Equation_e^x_(x_-_1)_=_e^-x_(x_+_1) | [
"Analysis"
] | [] | [
"Intermediate Value Theorem",
"Newton's Method"
] |
proofwiki-16604 | Mergelyan-Wesler Theorem | Let $P = \sequence {D_1, D_2, \dotsc}$ be an infinite sequence of disjoint open disks whose union is the unit disk $D$ except for a set of measure zero.
Let $r_n$ be the radius of $D_n$.
Then:
:$\ds \sum_{k \mathop = 1}^\infty r_k = +\infty$ | {{ProofWanted}}
{{Namedfor|Sergey Nikitovich Mergelyan|name2 = Oscar Wesler|cat = Mergelyan|cat2 = Wesler}} | Let $P = \sequence {D_1, D_2, \dotsc}$ be an [[Definition:Infinite Sequence|infinite sequence]] of [[Definition:Disjoint Sets|disjoint]] [[Definition:Open Complex Disk|open disks]] whose [[Definition:Set Union|union]] is the [[Definition:Unit Disk|unit disk]] $D$ except for a [[Definition:Set|set]] of [[Definition:Meas... | {{ProofWanted}}
{{Namedfor|Sergey Nikitovich Mergelyan|name2 = Oscar Wesler|cat = Mergelyan|cat2 = Wesler}} | Mergelyan-Wesler Theorem | https://proofwiki.org/wiki/Mergelyan-Wesler_Theorem | https://proofwiki.org/wiki/Mergelyan-Wesler_Theorem | [
"Complex Analysis"
] | [
"Definition:Sequence/Infinite Sequence",
"Definition:Disjoint Sets",
"Definition:Complex Disk/Open",
"Definition:Set Union",
"Definition:Unit Disk",
"Definition:Set",
"Definition:Null Set",
"Definition:Disk/Radius"
] | [] |
proofwiki-16605 | Integer Arbitrarily Close to Rational in Valuation Ring of P-adic Norm | Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime number $p$.
Let $x \in \Q$ such that $\norm x_p \le 1$.
Then for all $i \in \N$ there exists $\alpha \in \Z$ such that:
:$\norm {x - \alpha}_p \le p^{-i}$ | Let $i \in \N$.
Let $x = \dfrac a b: a, b \in \Z \text{ and } b \ne 0$.
{{WLOG}} we can assume that $\dfrac a b$ is in canonical form.
By Valuation Ring of P-adic Norm on Rationals:
:$\dfrac a b \in \Z_{\paren p} = \set {\dfrac c d \in \Q : p \nmid d}$
So $p \nmid b$.
Since $p \nmid b$, by Prime not Divisor implies Cop... | Let $\norm {\,\cdot\,}_p$ be the [[Definition:P-adic Norm|$p$-adic norm]] on the [[Definition:Rational Numbers|rationals $\Q$]] for some [[Definition:Prime Number|prime number]] $p$.
Let $x \in \Q$ such that $\norm x_p \le 1$.
Then for all $i \in \N$ there exists $\alpha \in \Z$ such that:
:$\norm {x - \alpha}_p \le... | Let $i \in \N$.
Let $x = \dfrac a b: a, b \in \Z \text{ and } b \ne 0$.
{{WLOG}} we can assume that $\dfrac a b$ is in [[Definition:Canonical Form of Rational Number|canonical form]].
By [[Valuation Ring of P-adic Norm on Rationals]]:
:$\dfrac a b \in \Z_{\paren p} = \set {\dfrac c d \in \Q : p \nmid d}$
So $p \nmi... | Integer Arbitrarily Close to Rational in Valuation Ring of P-adic Norm | https://proofwiki.org/wiki/Integer_Arbitrarily_Close_to_Rational_in_Valuation_Ring_of_P-adic_Norm | https://proofwiki.org/wiki/Integer_Arbitrarily_Close_to_Rational_in_Valuation_Ring_of_P-adic_Norm | [
"P-adic Number Theory"
] | [
"Definition:P-adic Norm",
"Definition:Rational Number",
"Definition:Prime Number"
] | [
"Definition:Rational Number/Canonical Form",
"Valuation Ring of P-adic Norm on Rationals",
"Prime not Divisor implies Coprime",
"Integer Combination of Coprime Integers"
] |
proofwiki-16606 | P-adic Integer has Unique Coherent Sequence Representative | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a$ be a $p$-adic number, that is left coset, in $\Q_p$ such that $\norm a _p \le 1$.
Then $a$ has exactly one representative that is a coherent sequence. | Let $\sequence{\beta_n}$ be a sequence representing $a$.
That is, $\sequence{\beta_n}$ is a Cauchy sequence in the $p$-adic norm $\norm{\,\cdot\,}_p$ on the rational numbers $\Q$.
By definition of a Cauchy sequence:
:$\forall j \in \N : \exists \mathop {\map M j} \in \N: \forall m, n \in \N: m, n \ge \map M j : \norm {... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $a$ be a [[Definition:P-adic Number|$p$-adic number]], that is [[Definition:Left Coset|left coset]], in $\Q_p$ such that $\norm a _p \le 1$.
The... | Let $\sequence{\beta_n}$ be a [[Definition:Sequence|sequence]] [[Definition:Representative of P-adic Number|representing]] $a$.
That is, $\sequence{\beta_n}$ is a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]] in the [[Definition:P-adic Norm|$p$-adic norm]] $\norm{\,\cdot\,}_p$ on the [[Definit... | P-adic Integer has Unique Coherent Sequence Representative | https://proofwiki.org/wiki/P-adic_Integer_has_Unique_Coherent_Sequence_Representative | https://proofwiki.org/wiki/P-adic_Integer_has_Unique_Coherent_Sequence_Representative | [
"P-adic Number Theory",
"P-adic Integer has Unique Coherent Sequence Representative"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:P-adic Number",
"Definition:Coset/Left Coset",
"Definition:P-adic Number/Representative",
"Definition:P-adically Coherent Sequence"
] | [
"Definition:Sequence",
"Definition:P-adic Number/Representative",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:P-adic Norm",
"Definition:Rational Number",
"Definition:Cauchy Sequence/Normed Division Ring",
"P-adic Norm of p-adic Number is Power of p",
"Definition:P-adic Norm",
"Def... |
proofwiki-16607 | P-adic Integer has Unique Coherent Sequence Representative/P-adic Expansion | Then $a$ has exactly one representative that is a $p$-adic expansion of the form:
:$\ds \sum_{n \mathop = 0}^\infty d_n p^n$ | From P-adic Integer has Unique Coherent Sequence Representative, $a$ has exactly one representative coherent sequence.
Let $\sequence{\alpha_n}$ be the coherent sequence such that:
:$\sequence{\alpha_n}$ is a representative of $a$.
From Coherent Sequence is Partial Sum of P-adic Expansion, there exists a $p$-adic expan... | Then $a$ has exactly one [[Definition:Representative of P-adic Number|representative]] that is a [[Definition:P-adic Expansion|$p$-adic expansion]] of the form:
:$\ds \sum_{n \mathop = 0}^\infty d_n p^n$ | From [[P-adic Integer has Unique Coherent Sequence Representative]], $a$ has exactly one [[Definition:Representative of P-adic Number|representative]] [[Definition:P-adically Coherent Sequence|coherent sequence]].
Let $\sequence{\alpha_n}$ be the [[Definition:P-adically Coherent Sequence|coherent sequence]] such that... | P-adic Integer has Unique Coherent Sequence Representative/P-adic Expansion | https://proofwiki.org/wiki/P-adic_Integer_has_Unique_Coherent_Sequence_Representative/P-adic_Expansion | https://proofwiki.org/wiki/P-adic_Integer_has_Unique_Coherent_Sequence_Representative/P-adic_Expansion | [
"P-adic Integer has Unique Coherent Sequence Representative"
] | [
"Definition:P-adic Number/Representative",
"Definition:P-adic Expansion"
] | [
"P-adic Integer has Unique Coherent Sequence Representative",
"Definition:P-adic Number/Representative",
"Definition:P-adically Coherent Sequence",
"Definition:P-adically Coherent Sequence",
"Definition:P-adic Number/Representative",
"Coherent Sequence is Partial Sum of P-adic Expansion",
"Definition:P-... |
proofwiki-16608 | P-adic Number has Unique P-adic Expansion Representative | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a$ be a $p$-adic number, that is left coset, in $\Q_p$.
Then $a$ has exactly one representative that is a $p$-adic expansion. | === Case 1 ===
Let $\norm a_p \le 1$.
From P-adic Integer has Unique P-adic Expansion Representative, $a$ has exactly one representative that is a $p$-adic expansion of the form:
:$\ds \sum_{n \mathop = 0}^\infty d_n p^n$
{{qed|lemma}} | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $a$ be a [[Definition:P-adic Number|$p$-adic number]], that is [[Definition:Left Coset|left coset]], in $\Q_p$.
Then $a$ has exactly one [[Defin... | === Case 1 ===
Let $\norm a_p \le 1$.
From [[P-adic Integer has Unique P-adic Expansion Representative]], $a$ has exactly one [[Definition:Representative of P-adic Number|representative]] that is a [[Definition:P-adic Expansion|$p$-adic expansion]] of the form:
:$\ds \sum_{n \mathop = 0}^\infty d_n p^n$
{{qed|lemma}} | P-adic Number has Unique P-adic Expansion Representative | https://proofwiki.org/wiki/P-adic_Number_has_Unique_P-adic_Expansion_Representative | https://proofwiki.org/wiki/P-adic_Number_has_Unique_P-adic_Expansion_Representative | [
"P-adic Number Theory"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:P-adic Number",
"Definition:Coset/Left Coset",
"Definition:P-adic Number/Representative",
"Definition:P-adic Expansion"
] | [
"P-adic Integer has Unique Coherent Sequence Representative/P-adic Expansion",
"Definition:P-adic Number/Representative",
"Definition:P-adic Expansion",
"P-adic Integer has Unique Coherent Sequence Representative/P-adic Expansion",
"Definition:P-adic Number/Representative",
"Definition:P-adic Expansion",
... |
proofwiki-16609 | Value of Plastic Constant | The plastic constant $P$ is evaluated as:
{{begin-eqn}}
{{eqn | l = P
| r = \sqrt [3] {\frac {9 + \sqrt {69} } {18} } + \sqrt [3] {\frac {9 - \sqrt {69} } {18} }
| c =
}}
{{eqn | r = 1 \cdotp 32471 \, 79572 \, 44746 \, 02596 \, 09088 \, 54 \ldots
| c =
}}
{{end-eqn}} | By definition, the plastic constant $P$ is the real root of the cubic:
:$x^3 - x - 1 = 0$
Recall Cardano's Formula:
{{:Cardano's Formula}}
Here we have:
{{begin-eqn}}
{{eqn | l = a
| r = 1
}}
{{eqn | l = b
| r = 0
}}
{{eqn | l = c
| r = -1
}}
{{eqn | l = d
| r = -1
}}
{{end-eqn}}
Hence:
{{begin-... | The [[Definition:Plastic Constant|plastic constant]] $P$ is evaluated as:
{{begin-eqn}}
{{eqn | l = P
| r = \sqrt [3] {\frac {9 + \sqrt {69} } {18} } + \sqrt [3] {\frac {9 - \sqrt {69} } {18} }
| c =
}}
{{eqn | r = 1 \cdotp 32471 \, 79572 \, 44746 \, 02596 \, 09088 \, 54 \ldots
| c =
}}
{{end-eqn}} | By definition, the [[Definition:Plastic Constant|plastic constant]] $P$ is the [[Definition:Real Number|real]] [[Definition:Root of Polynomial|root]] of the [[Definition:Cubic Equation|cubic]]:
:$x^3 - x - 1 = 0$
Recall [[Cardano's Formula]]:
{{:Cardano's Formula}}
Here we have:
{{begin-eqn}}
{{eqn | l = a
| r... | Value of Plastic Constant | https://proofwiki.org/wiki/Value_of_Plastic_Constant | https://proofwiki.org/wiki/Value_of_Plastic_Constant | [
"Pisot-Vijayaraghavan Numbers"
] | [
"Definition:Plastic Constant"
] | [
"Definition:Plastic Constant",
"Definition:Real Number",
"Definition:Root of Polynomial",
"Definition:Cubic Equation",
"Cardano's Formula",
"Definition:Root of Polynomial",
"Definition:Zero (Number)",
"Definition:Complex Number/Imaginary Part",
"Definition:Root of Polynomial",
"Definition:Real Num... |
proofwiki-16610 | Unique Integer Close to Rational in Valuation Ring of P-adic Norm | Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime number $p$.
Let $x \in \Q$ such that $\norm{x}_p \le 1$.
Then for all $i \in \N$ there exists a unique $\alpha \in \Z$ such that:
:$(1): \quad \norm {x - \alpha}_p \le p^{-i}$
:$(2): \quad 0 \le \alpha \le p^i - 1$ | Let $i \in \N$.
From Integer Arbitrarily Close to Rational in Valuation Ring of P-adic Norm:
:$\exists \mathop {\alpha'} \in \Z: \norm{x - \alpha'}_p \le p^{-i}$
By Integer is Congruent to Integer less than Modulus, then there exists $\alpha \in \Z$:
:$\alpha \equiv \alpha' \pmod {p^i}$.
:$0 \le \alpha \le p^i - 1$
The... | Let $\norm {\,\cdot\,}_p$ be the [[Definition:P-adic Norm|$p$-adic norm]] on the [[Definition:Rational Numbers|rationals $\Q$]] for some [[Definition:Prime Number|prime number]] $p$.
Let $x \in \Q$ such that $\norm{x}_p \le 1$.
Then for all $i \in \N$ there exists a [[Definition:Unique|unique]] $\alpha \in \Z$ such ... | Let $i \in \N$.
From [[Integer Arbitrarily Close to Rational in Valuation Ring of P-adic Norm]]:
:$\exists \mathop {\alpha'} \in \Z: \norm{x - \alpha'}_p \le p^{-i}$
By [[Integer is Congruent to Integer less than Modulus]], then there exists $\alpha \in \Z$:
:$\alpha \equiv \alpha' \pmod {p^i}$.
:$0 \le \alpha \le ... | Unique Integer Close to Rational in Valuation Ring of P-adic Norm | https://proofwiki.org/wiki/Unique_Integer_Close_to_Rational_in_Valuation_Ring_of_P-adic_Norm | https://proofwiki.org/wiki/Unique_Integer_Close_to_Rational_in_Valuation_Ring_of_P-adic_Norm | [
"P-adic Number Theory"
] | [
"Definition:P-adic Norm",
"Definition:Rational Number",
"Definition:Prime Number",
"Definition:Unique"
] | [
"Integer Arbitrarily Close to Rational in Valuation Ring of P-adic Norm",
"Integer is Congruent to Integer less than Modulus",
"Properties of Norm on Division Ring/Norm of Negative",
"Initial Segment of Natural Numbers forms Complete Residue System"
] |
proofwiki-16611 | Hermite Constant for Dimension 2 | The Hermite constant for dimension $2$ is:
:$\gamma_2 = \dfrac 2 {\sqrt 3}$
or, as it is often presented:
:$\paren {\gamma_2}^2 = \dfrac 4 3$ | The statement of the result to be proved can be expressed as:
:There exist non-zero $x$ and $y$ such that:
::$\paren {a x^2 + 2 b x y + c y^2}^2 \le \size {a c - b^2} \times \dfrac 4 3$
{{ProofWanted|over my head}} | The [[Definition:Hermite Constant|Hermite constant]] for dimension $2$ is:
:$\gamma_2 = \dfrac 2 {\sqrt 3}$
or, as it is often presented:
:$\paren {\gamma_2}^2 = \dfrac 4 3$ | The statement of the result to be proved can be expressed as:
:There exist non-zero $x$ and $y$ such that:
::$\paren {a x^2 + 2 b x y + c y^2}^2 \le \size {a c - b^2} \times \dfrac 4 3$
{{ProofWanted|over my head}} | Hermite Constant for Dimension 2 | https://proofwiki.org/wiki/Hermite_Constant_for_Dimension_2 | https://proofwiki.org/wiki/Hermite_Constant_for_Dimension_2 | [
"Hermite Constants"
] | [
"Definition:Hermite Constant"
] | [] |
proofwiki-16612 | Angle between Straight Lines in Plane | Let $L_1$ and $L_2$ be straight lines embedded in a cartesian plane, given by the equations:
{{begin-eqn}}
{{eqn | q = L_1
| l = y
| r = m_1 x + c_1
}}
{{eqn | q = L_2
| l = y
| r = m_2 x + c_2
}}
{{end-eqn}}
Then the angle $\psi$ between $L_1$ and $L_2$ is given by:
:$\psi = \arctan \dfrac {m_1... | :500px
Let $\psi_1$ and $\psi_2$ be the angles that $L_1$ and $L_2$ make with the $x$-axis respectively.
Then by the definition of slope:
{{begin-eqn}}
{{eqn | l = \tan \psi_1
| r = m_1
}}
{{eqn | l = \tan \psi_2
| r = m_2
}}
{{end-eqn}}
and so:
{{begin-eqn}}
{{eqn | l = \tan \psi
| r = \map \tan {\ps... | Let $L_1$ and $L_2$ be [[Definition:Straight Line|straight lines]] embedded in a [[Definition:Cartesian Plane|cartesian plane]], given by the [[Equation of Straight Line in Plane/Slope-Intercept Form|equations]]:
{{begin-eqn}}
{{eqn | q = L_1
| l = y
| r = m_1 x + c_1
}}
{{eqn | q = L_2
| l = y
... | :[[File:Angle-between-Straight-Lines.png|500px]]
Let $\psi_1$ and $\psi_2$ be the [[Definition:Angle|angles]] that $L_1$ and $L_2$ make with the [[Definition:X-Axis|$x$-axis]] respectively.
Then by the definition of [[Definition:Slope of Straight Line|slope]]:
{{begin-eqn}}
{{eqn | l = \tan \psi_1
| r = m_1
}... | Angle between Straight Lines in Plane | https://proofwiki.org/wiki/Angle_between_Straight_Lines_in_Plane | https://proofwiki.org/wiki/Angle_between_Straight_Lines_in_Plane | [
"Angle between Straight Lines in Plane",
"Straight Lines",
"Angles"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Equation of Straight Line in Plane/Slope-Intercept Form",
"Definition:Angle"
] | [
"File:Angle-between-Straight-Lines.png",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Definition:Slope/Straight Line",
"Tangent of Difference"
] |
proofwiki-16613 | Slope of Lemniscate of Bernoulli at Origin | Consider the lemniscate of Bernoulli $M$ embedded in a Cartesian plane such that its foci are at $\tuple {a, 0}$ and $\tuple {-a, 0}$ respectively.
Let $O$ denote the origin.
The tangents to $M$ at $O$ are at an angle of $45 \degrees = \dfrac \pi 4$ to the $x$-axis.
:630px | Recall the parametric definition of lemniscate of Bernoulli:
:$\begin {cases} x = \dfrac {a \sqrt 2 \cos t} {\sin^2 t + 1} \\ \\ y = \dfrac {a \sqrt 2 \cos t \sin t} {\sin^2 t + 1} \end {cases}$
Note that $\tuple {x, y} = \tuple {0, 0}$ {{iff}} $\cos t = 0$.
At these points we have $\sin t = \pm \sqrt {1 - \cos^2 t} = ... | Consider the [[Definition:Lemniscate of Bernoulli|lemniscate of Bernoulli]] $M$ embedded in a [[Definition:Cartesian Plane|Cartesian plane]] such that its [[Definition:Focus of Lemniscate of Bernoulli|foci]] are at $\tuple {a, 0}$ and $\tuple {-a, 0}$ respectively.
Let $O$ denote the [[Definition:Origin|origin]].
Th... | Recall the [[Definition:Parametric Equation|parametric]] definition of [[Definition:Lemniscate of Bernoulli/Parametric Definition|lemniscate of Bernoulli]]:
:$\begin {cases} x = \dfrac {a \sqrt 2 \cos t} {\sin^2 t + 1} \\ \\ y = \dfrac {a \sqrt 2 \cos t \sin t} {\sin^2 t + 1} \end {cases}$
Note that $\tuple {x, y} = \... | Slope of Lemniscate of Bernoulli at Origin | https://proofwiki.org/wiki/Slope_of_Lemniscate_of_Bernoulli_at_Origin | https://proofwiki.org/wiki/Slope_of_Lemniscate_of_Bernoulli_at_Origin | [
"Lemniscate of Bernoulli"
] | [
"Definition:Lemniscate of Bernoulli",
"Definition:Cartesian Plane",
"Definition:Lemniscate of Bernoulli/Focus",
"Definition:Coordinate System/Origin",
"Definition:Tangent Line",
"Definition:Angle",
"Definition:Axis/X-Axis",
"File:Lemniscate-tangents-at-origin.png"
] | [
"Definition:Parametric Equation",
"Definition:Lemniscate of Bernoulli/Parametric Definition",
"Quotient Rule for Derivatives",
"Quotient Rule for Derivatives",
"Derivative of Composite Function",
"Definition:Tangent Line",
"Definition:Angle",
"Definition:Axis/X-Axis"
] |
proofwiki-16614 | Area of Lobe of Lemniscate of Bernoulli | Consider the lemniscate of Bernoulli $M$ embedded in a Cartesian plane such that its foci are at $\tuple {a, 0}$ and $\tuple {-a, 0}$ respectively.
Let $O$ denote the origin.
The area of one lobe of $M$ is $a^2$. | By the definition of the lemniscate of Bernoulli, we have that the polar equation of $M$ is:
:$r^2 = 2 a^2 \cos 2 \theta$
Let $\AA$ denote the area of one lobe of $M$.
The boundary of the right hand lobe of $M$ is traced out where $-\dfrac \pi 2 \le 2 \theta \le \dfrac \pi 2$.
Thus:
{{begin-eqn}}
{{eqn | l = \AA
... | Consider the [[Definition:Lemniscate of Bernoulli|lemniscate of Bernoulli]] $M$ embedded in a [[Definition:Cartesian Plane|Cartesian plane]] such that its [[Definition:Focus of Lemniscate of Bernoulli|foci]] are at $\tuple {a, 0}$ and $\tuple {-a, 0}$ respectively.
Let $O$ denote the [[Definition:Origin|origin]].
Th... | By the definition of the [[Definition:Lemniscate of Bernoulli/Polar Definition|lemniscate of Bernoulli]], we have that the [[Definition:Polar Equation|polar equation]] of $M$ is:
:$r^2 = 2 a^2 \cos 2 \theta$
Let $\AA$ denote the [[Definition:Area|area]] of one [[Definition:Lobe of Lemniscate of Bernoulli|lobe]] of $M... | Area of Lobe of Lemniscate of Bernoulli | https://proofwiki.org/wiki/Area_of_Lobe_of_Lemniscate_of_Bernoulli | https://proofwiki.org/wiki/Area_of_Lobe_of_Lemniscate_of_Bernoulli | [
"Lemniscate of Bernoulli"
] | [
"Definition:Lemniscate of Bernoulli",
"Definition:Cartesian Plane",
"Definition:Lemniscate of Bernoulli/Focus",
"Definition:Coordinate System/Origin",
"Definition:Area",
"Definition:Lemniscate of Bernoulli/Lobe"
] | [
"Definition:Lemniscate of Bernoulli/Polar Definition",
"Definition:Polar Equation",
"Definition:Area",
"Definition:Lemniscate of Bernoulli/Lobe",
"Definition:Lemniscate of Bernoulli/Lobe",
"Area between Radii and Curve in Polar Coordinates",
"Primitive of Cosine Function/Corollary",
"Sine of Right Ang... |
proofwiki-16615 | Equation of Cardioid/Polar | Let $C$ be a cardioid embedded in a polar coordinate plane such that:
:its deferent of radius $a$ is positioned with its center at $\polar {a, 0}$
:there is a cusp at the origin.
The polar equation of $C$ is:
:$r = 2 a \paren {1 + \cos \theta}$ | :525px
Let $P = \polar {r, \theta}$ be an arbitrary point on $C$.
Let $A$ and $B$ be the centers of the deferent and epicycle respectively.
Let $Q$ be the point where the deferent and epicycle touch.
By definition of the method of construction of $C$, we have that the arc $OQ$ of the deferent equals the arc $PQ$ of the... | Let $C$ be a [[Definition:Cardioid|cardioid]] embedded in a [[Definition:Polar Coordinate Plane|polar coordinate plane]] such that:
:its [[Definition:Deferent of Epicycloid|deferent]] of [[Definition:Radius of Circle|radius]] $a$ is positioned with its [[Definition:Center of Circle|center]] at $\polar {a, 0}$
:there is... | :[[File:Cardioid-right-construction.png|525px]]
Let $P = \polar {r, \theta}$ be an arbitrary [[Definition:Point|point]] on $C$.
Let $A$ and $B$ be the [[Definition:Center of Circle|centers]] of the [[Definition:Deferent of Epicycloid|deferent]] and [[Definition:Epicycle of Epicycloid|epicycle]] respectively.
Let $Q... | Equation of Cardioid/Polar | https://proofwiki.org/wiki/Equation_of_Cardioid/Polar | https://proofwiki.org/wiki/Equation_of_Cardioid/Polar | [
"Cardioids"
] | [
"Definition:Cardioid",
"Definition:Polar Coordinates/Polar Plane",
"Definition:Epicycloid/Generator/Deferent",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Epicycloid/Cusp",
"Definition:Coordinate System/Origin",
"Definition:Polar Equation"
] | [
"File:Cardioid-right-construction.png",
"Definition:Point",
"Definition:Circle/Center",
"Definition:Epicycloid/Generator/Deferent",
"Definition:Epicycloid/Generator/Epicycle",
"Definition:Point",
"Definition:Epicycloid/Generator/Deferent",
"Definition:Epicycloid/Generator/Epicycle",
"Definition:Circ... |
proofwiki-16616 | Equation of Cardioid/Parametric | Let $C$ be a cardioid embedded in a Cartesian coordinate plane such that:
:its deferent of radius $a$ is positioned with its center at $\tuple {a, 0}$
:there is a cusp at the origin.
Then $C$ can be expressed by the parametric equation:
:$\begin {cases} x = 2 a \cos t \paren {1 + \cos t} \\ y = 2 a \sin t \paren {1 + \... | :525px
Let $P = \polar {x, y}$ be an arbitrary point on $C$.
From Polar Form of Equation of Cardioid, $C$ is expressed as a polar equation as:
:$r = 2 a \paren {1 + \cos \theta}$
We have that:
:$x = r \cos \theta$
:$y = r \sin \theta$
Replace $\theta$ with $t$ and the required parametric equation is the result.
{{qed}} | Let $C$ be a [[Definition:Cardioid|cardioid]] embedded in a [[Definition:Cartesian Plane|Cartesian coordinate plane]] such that:
:its [[Definition:Deferent of Epicycloid|deferent]] of [[Definition:Radius of Circle|radius]] $a$ is positioned with its [[Definition:Center of Circle|center]] at $\tuple {a, 0}$
:there is a ... | :[[File:Cardioid-right-construction.png|525px]]
Let $P = \polar {x, y}$ be an arbitrary [[Definition:Point|point]] on $C$.
From [[Equation of Cardioid/Polar|Polar Form of Equation of Cardioid]], $C$ is expressed as a [[Definition:Polar Equation|polar equation]] as:
:$r = 2 a \paren {1 + \cos \theta}$
We have that:... | Equation of Cardioid/Parametric | https://proofwiki.org/wiki/Equation_of_Cardioid/Parametric | https://proofwiki.org/wiki/Equation_of_Cardioid/Parametric | [
"Cardioids"
] | [
"Definition:Cardioid",
"Definition:Cartesian Plane",
"Definition:Epicycloid/Generator/Deferent",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Epicycloid/Cusp",
"Definition:Coordinate System/Origin",
"Definition:Parametric Equation"
] | [
"File:Cardioid-right-construction.png",
"Definition:Point",
"Equation of Cardioid/Polar",
"Definition:Polar Equation",
"Definition:Parametric Equation"
] |
proofwiki-16617 | Area inside Cardioid | Consider the cardioid $C$ embedded in a polar plane given by its polar equation:
:$r = 2 a \paren {1 + \cos \theta}$
The area inside $C$ is $6 \pi a^2$. | Let $\AA$ denote the area inside $C$.
The boundary of $C$ is traced out where $-\pi \le \theta \le \pi$.
Thus:
{{begin-eqn}}
{{eqn | l = \AA
| r = \int_{-\pi}^\pi \dfrac {\map {r^2} \theta} 2 \rd \theta
| c = Area between Radii and Curve in Polar Coordinates
}}
{{eqn | r = \int_{-\pi}^\pi \dfrac {\paren {2 ... | Consider the [[Definition:Cardioid|cardioid]] $C$ embedded in a [[Definition:Polar Coordinate Plane|polar plane]] given by its [[Equation of Cardioid/Polar|polar equation]]:
:$r = 2 a \paren {1 + \cos \theta}$
The [[Definition:Area|area]] inside $C$ is $6 \pi a^2$. | Let $\AA$ denote the [[Definition:Area|area]] inside $C$.
The boundary of $C$ is traced out where $-\pi \le \theta \le \pi$.
Thus:
{{begin-eqn}}
{{eqn | l = \AA
| r = \int_{-\pi}^\pi \dfrac {\map {r^2} \theta} 2 \rd \theta
| c = [[Area between Radii and Curve in Polar Coordinates]]
}}
{{eqn | r = \int_{... | Area inside Cardioid | https://proofwiki.org/wiki/Area_inside_Cardioid | https://proofwiki.org/wiki/Area_inside_Cardioid | [
"Cardioids",
"Area Formulas"
] | [
"Definition:Cardioid",
"Definition:Polar Coordinates/Polar Plane",
"Equation of Cardioid/Polar",
"Definition:Area"
] | [
"Definition:Area",
"Area between Radii and Curve in Polar Coordinates",
"Equation of Cardioid/Polar",
"Primitive of Cosine Function/Corollary",
"Primitive of Square of Cosine Function",
"Sine of Integer Multiple of Pi"
] |
proofwiki-16618 | Length of Perimeter of Cardioid | Consider the cardioid $C$ embedded in a polar plane given by its polar equation:
:$r = 2 a \paren {1 + \cos \theta}$
where $a > 0$.
The length of the perimeter of $C$ is $16 a$. | Let $\LL$ denote the length of the perimeter of $C$.
The boundary of the $C$ is traced out where $-\pi \le \theta \le \pi$.
From Arc Length for Parametric Equations:
:$\ds \LL = \int_{\theta \mathop = -\pi}^{\theta \mathop = \pi} \sqrt {\paren {\frac {\d x} {\d \theta} }^2 + \paren {\frac {\d y} {\d \theta} }^2} \rd \t... | Consider the [[Definition:Cardioid|cardioid]] $C$ embedded in a [[Definition:Polar Coordinate Plane|polar plane]] given by its [[Equation of Cardioid/Polar|polar equation]]:
:$r = 2 a \paren {1 + \cos \theta}$
where $a > 0$.
The [[Definition:Arc Length|length]] of the [[Definition:Perimeter|perimeter]] of $C$ is $16... | Let $\LL$ denote the [[Definition:Arc Length|length]] of the [[Definition:Perimeter|perimeter]] of $C$.
The boundary of the $C$ is traced out where $-\pi \le \theta \le \pi$.
From [[Arc Length for Parametric Equations]]:
:$\ds \LL = \int_{\theta \mathop = -\pi}^{\theta \mathop = \pi} \sqrt {\paren {\frac {\d x} {\d... | Length of Perimeter of Cardioid/Proof 1 | https://proofwiki.org/wiki/Length_of_Perimeter_of_Cardioid | https://proofwiki.org/wiki/Length_of_Perimeter_of_Cardioid/Proof_1 | [
"Length of Perimeter of Cardioid",
"Cardioids"
] | [
"Definition:Cardioid",
"Definition:Polar Coordinates/Polar Plane",
"Equation of Cardioid/Polar",
"Definition:Arc Length",
"Definition:Perimeter"
] | [
"Definition:Arc Length",
"Definition:Perimeter",
"Arc Length for Parametric Equations",
"Equation of Cardioid/Parametric",
"Derivative of Cosine Function",
"Derivative of Composite Function",
"Double Angle Formulas/Sine",
"Derivative of Sine Function",
"Product Rule for Derivatives",
"Double Angle... |
proofwiki-16619 | Length of Perimeter of Cardioid | Consider the cardioid $C$ embedded in a polar plane given by its polar equation:
:$r = 2 a \paren {1 + \cos \theta}$
where $a > 0$.
The length of the perimeter of $C$ is $16 a$. | Let $\LL$ denote the length of the perimeter of $C$.
The boundary of the $C$ is traced out where $-\pi \le \theta \le \pi$.
From Arc Length for Polar Curve:
:$\ds \LL = \int_{\theta \mathop = -\pi}^{\theta \mathop = \pi} \sqrt {r^2 + \paren {\frac {\d r} {\d \theta} }^2} \rd \theta$
where:
:$r = 2 a \paren {1 + \cos \t... | Consider the [[Definition:Cardioid|cardioid]] $C$ embedded in a [[Definition:Polar Coordinate Plane|polar plane]] given by its [[Equation of Cardioid/Polar|polar equation]]:
:$r = 2 a \paren {1 + \cos \theta}$
where $a > 0$.
The [[Definition:Arc Length|length]] of the [[Definition:Perimeter|perimeter]] of $C$ is $16... | Let $\LL$ denote the [[Definition:Arc Length|length]] of the [[Definition:Perimeter|perimeter]] of $C$.
The boundary of the $C$ is traced out where $-\pi \le \theta \le \pi$.
From [[Arc Length for Polar Curve]]:
:$\ds \LL = \int_{\theta \mathop = -\pi}^{\theta \mathop = \pi} \sqrt {r^2 + \paren {\frac {\d r} {\d \t... | Length of Perimeter of Cardioid/Proof 2 | https://proofwiki.org/wiki/Length_of_Perimeter_of_Cardioid | https://proofwiki.org/wiki/Length_of_Perimeter_of_Cardioid/Proof_2 | [
"Length of Perimeter of Cardioid",
"Cardioids"
] | [
"Definition:Cardioid",
"Definition:Polar Coordinates/Polar Plane",
"Equation of Cardioid/Polar",
"Definition:Arc Length",
"Definition:Perimeter"
] | [
"Definition:Arc Length",
"Definition:Perimeter",
"Arc Length for Polar Curve",
"Sum Rule for Derivatives",
"Derivative of Cosine Function",
"Sum of Squares of Sine and Cosine",
"Definite Integral of Even Function",
"Primitive of Cosine Function/Corollary",
"Sine of Right Angle",
"Sine of Zero is Z... |
proofwiki-16620 | Arc Length for Polar Curve | Let $a$ and $b$ be real numbers.
Let $\CC$ be a simple curve continuous on $\closedint a b$ and continuously differentiable on $\openint a b$.
Let $\CC$ be described by the parametric equations:
:$\begin {cases} x & = r \cos \theta \\ y & = r \sin \theta \end {cases}$
where:
:$r$ is a function of $\theta$
:$\theta \in ... | Note that $\CC$ satisfies the conditions of Arc Length for Parametric Equations.
So:
:$\ds s = \int_a^b \sqrt {\paren {\frac {\d x} {\d \theta} }^2 + \paren {\frac {\d y} {\d \theta} }^2} \rd \theta$
We have:
{{begin-eqn}}
{{eqn | l = \frac {\d x} {\d \theta}
| r = \frac \d {\d \theta} \paren {r \cos \theta}
}}
{{eq... | Let $a$ and $b$ be [[Definition:Real Number|real numbers]].
Let $\CC$ be a [[Definition:Simple Contour|simple]] [[Definition:Curve|curve]] [[Definition:Continuous on Interval|continuous]] on $\closedint a b$ and [[Definition:Continuously Differentiable|continuously differentiable]] on $\openint a b$.
Let $\CC$ be des... | Note that $\CC$ satisfies the conditions of [[Arc Length for Parametric Equations]].
So:
:$\ds s = \int_a^b \sqrt {\paren {\frac {\d x} {\d \theta} }^2 + \paren {\frac {\d y} {\d \theta} }^2} \rd \theta$
We have:
{{begin-eqn}}
{{eqn | l = \frac {\d x} {\d \theta}
| r = \frac \d {\d \theta} \paren {r \cos \theta}... | Arc Length for Polar Curve | https://proofwiki.org/wiki/Arc_Length_for_Polar_Curve | https://proofwiki.org/wiki/Arc_Length_for_Polar_Curve | [
"Integral Calculus"
] | [
"Definition:Real Number",
"Definition:Contour/Simple",
"Definition:Line/Curve",
"Definition:Continuous Real Function/Interval",
"Definition:Continuously Differentiable",
"Definition:Parametric Equation",
"Definition:Function",
"Definition:Arc Length"
] | [
"Arc Length for Parametric Equations",
"Product Rule for Derivatives",
"Derivative of Cosine Function",
"Product Rule for Derivatives",
"Derivative of Sine Function",
"Square of Sum",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-16621 | Number of Petals of Odd Index Rhodonea Curve | Let $n$ be an odd positive integer.
Let $R$ be a '''rhodonea curve''' defined by one of the polar equations:
{{begin-eqn}}
{{eqn | l = r
| r = a \cos n \theta
}}
{{eqn | l = r
| r = a \sin n \theta
}}
{{end-eqn}}
Then $R$ has $n$ petals. | Let $R$ be defined by $r = a \cos n \theta$.
The tips of each of the petals of $R$ occur when $\cos n \theta = \pm 1$.
This happens whenever $n \theta \in \set {0, \pi}$.
Thus for $0 \le \theta < 2 \pi$, the tips occur at:
:$\theta \in \set {\dfrac {2 k \pi} {2 n}: k \in \set {0, 1, \ldots, 2 n - 1} }$
Let $k < n$.
The... | Let $n$ be an [[Definition:Odd Integer|odd]] [[Definition:Positive Integer|positive integer]].
Let $R$ be a '''[[Definition:Rhodonea Curve|rhodonea curve]]''' defined by one of the [[Definition:Polar Equation|polar equations]]:
{{begin-eqn}}
{{eqn | l = r
| r = a \cos n \theta
}}
{{eqn | l = r
| r = a \si... | Let $R$ be defined by $r = a \cos n \theta$.
The tips of each of the [[Definition:Petal of Rhodonea Curve|petals]] of $R$ occur when $\cos n \theta = \pm 1$.
This happens whenever $n \theta \in \set {0, \pi}$.
Thus for $0 \le \theta < 2 \pi$, the tips occur at:
:$\theta \in \set {\dfrac {2 k \pi} {2 n}: k \in \set {... | Number of Petals of Odd Index Rhodonea Curve | https://proofwiki.org/wiki/Number_of_Petals_of_Odd_Index_Rhodonea_Curve | https://proofwiki.org/wiki/Number_of_Petals_of_Odd_Index_Rhodonea_Curve | [
"Rhodonea Curves"
] | [
"Definition:Odd Integer",
"Definition:Positive/Integer",
"Definition:Rhodonea Curve",
"Definition:Polar Equation",
"Definition:Rhodonea Curve/Petal"
] | [
"Definition:Rhodonea Curve/Petal",
"Definition:Distinct/Plural",
"Definition:Rhodonea Curve/Petal",
"Cosine of Angle plus Straight Angle",
"Definition:Rhodonea Curve/Petal"
] |
proofwiki-16622 | Equation of Trochoid | Consider a circle $C$ of radius $a$ rolling without slipping along the x-axis of a cartesian plane.
Consider the point $P$ on on the line of a radius of $C$ at a distance $b$ from the center of $C$.
Let $P$ be on the y-axis when the center of $C$ is also on the y-axis.
Consider the trochoid traced out by the point $P$.... | Let $C$ have rolled so that the radius to the point $P = \tuple {x, y}$ is at angle $\theta$ to the vertical.
The center of $C$ is at $\tuple {a \theta, a}$.
Then it follows from the definition of sine and cosine that:
:$x = a \theta - b \sin \theta$
:$y = a - b \cos \theta$
whence the result.
{{qed}} | Consider a [[Definition:Circle|circle]] $C$ of [[Definition:Radius of Circle|radius]] $a$ rolling without slipping along the [[Definition:X-Axis|x-axis]] of a [[Definition:Cartesian Plane|cartesian plane]].
Consider the [[Definition:Point|point]] $P$ on on the line of a [[Definition:Radius of Circle|radius]] of $C$ at... | Let $C$ have rolled so that the [[Definition:Radius of Circle|radius]] to the point $P = \tuple {x, y}$ is at [[Definition:Angle|angle]] $\theta$ to [[Definition:Vertical|the vertical]].
The [[Definition:Center of Circle|center]] of $C$ is at $\tuple {a \theta, a}$.
Then it follows from the definition of [[Definitio... | Equation of Trochoid | https://proofwiki.org/wiki/Equation_of_Trochoid | https://proofwiki.org/wiki/Equation_of_Trochoid | [
"Trochoids"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Axis/X-Axis",
"Definition:Cartesian Plane",
"Definition:Point",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Axis/Y-Axis",
"Definition:Circle/Center",
"Definition:Axis/Y-Axis",
"Definition:Trochoid",
"Definitio... | [
"Definition:Circle/Radius",
"Definition:Angle",
"Definition:Vertical",
"Definition:Circle/Center",
"Definition:Sine",
"Definition:Cosine"
] |
proofwiki-16623 | Equation of Witch of Agnesi/Parametric Form | The equation of the Witch of Agnesi can be presented in parametric form as:
:$\begin {cases} x = 2 a \cot \theta \\ y = a \paren {1 - \cos 2 \theta} \end {cases}$ | Let $P = \tuple {x, y}$ and $A = \tuple {d, y}$.
Let $\theta$ be the angle that $ON$ makes with the horizontal.
We have by definition of cotangent:
:$\dfrac {OM} {MN} = \dfrac {2 a} x = \cot \theta$
By Thales' Theorem $\angle OAM$ is a right angle.
Hence $\angle OMA = \theta$ and so:
:$OA = 2 a \cos \theta$
Thus:
:$2 a... | The equation of the [[Definition:Witch of Agnesi|Witch of Agnesi]] can be presented in [[Definition:Parametric Equation|parametric form]] as:
:$\begin {cases} x = 2 a \cot \theta \\ y = a \paren {1 - \cos 2 \theta} \end {cases}$ | Let $P = \tuple {x, y}$ and $A = \tuple {d, y}$.
Let $\theta$ be the [[Definition:Angle|angle]] that $ON$ makes with the [[Definition:Horizontal|horizontal]].
We have by definition of [[Definition:Cotangent of Angle|cotangent]]:
:$\dfrac {OM} {MN} = \dfrac {2 a} x = \cot \theta$
By [[Thales' Theorem]] $\angle OAM$ ... | Equation of Witch of Agnesi/Parametric Form | https://proofwiki.org/wiki/Equation_of_Witch_of_Agnesi/Parametric_Form | https://proofwiki.org/wiki/Equation_of_Witch_of_Agnesi/Parametric_Form | [
"Witch of Agnesi"
] | [
"Definition:Witch of Agnesi",
"Definition:Parametric Equation"
] | [
"Definition:Angle",
"Definition:Horizontal",
"Definition:Cotangent/Definition from Triangle",
"Thales' Theorem",
"Definition:Right Angle"
] |
proofwiki-16624 | Area of Loop of Folium of Descartes | Consider the folium of Descartes $F$, given in parametric form as:
:$\begin {cases} x = \dfrac {3 a t} {1 + t^3} \\ y = \dfrac {3 a t^2} {1 + t^3} \end {cases}$
The area $\AA$ of the loop of $F$ is given as:
:$\AA = \dfrac {3 a^2} 2$ | From Behaviour of Parametric Equations for Folium of Descartes according to Parameter we have that the loop is traversed for $0 \le t < +\infty$.
We convert the parametric equation to polar form:
{{begin-eqn}}
{{eqn | l = r^2
| r = x^2 + y^2
| c =
}}
{{eqn | r = \dfrac {\paren {3 a t}^2} {\paren {1 + t^3}^... | Consider the [[Definition:Folium of Descartes|folium of Descartes]] $F$, given in [[Definition:Folium of Descartes/Parametric Form|parametric form]] as:
:$\begin {cases} x = \dfrac {3 a t} {1 + t^3} \\ y = \dfrac {3 a t^2} {1 + t^3} \end {cases}$
The [[Definition:Area|area]] $\AA$ of the loop of $F$ is given as:
:$\... | From [[Behaviour of Parametric Equations for Folium of Descartes according to Parameter]] we have that the loop is traversed for $0 \le t < +\infty$.
We convert the [[Definition:Parametric Equation|parametric equation]] to [[Definition:Polar Equation|polar form]]:
{{begin-eqn}}
{{eqn | l = r^2
| r = x^2 + y^2
... | Area of Loop of Folium of Descartes | https://proofwiki.org/wiki/Area_of_Loop_of_Folium_of_Descartes | https://proofwiki.org/wiki/Area_of_Loop_of_Folium_of_Descartes | [
"Folium of Descartes"
] | [
"Definition:Folium of Descartes",
"Definition:Folium of Descartes/Parametric Form",
"Definition:Area"
] | [
"Behaviour of Parametric Equations for Folium of Descartes according to Parameter",
"Definition:Parametric Equation",
"Definition:Polar Equation",
"Derivative of Arctangent Function",
"Area between Radii and Curve in Polar Coordinates",
"Integration by Substitution",
"Integration by Substitution",
"In... |
proofwiki-16625 | Asymptote to Folium of Descartes | Consider the folium of Descartes $F$, given in parametric form as:
:$\begin {cases} x = \dfrac {3 a t} {1 + t^3} \\ y = \dfrac {3 a t^2} {1 + t^3} \end {cases}$
The straight line whose equation is given by:
:$x + y + a = 0$
is an asymptote to $F$. | First we note that from Behaviour of Parametric Equations for Folium of Descartes according to Parameter:
:when $t = 0$ we have that $x = y = 0$
:when $t \to \pm \infty$ we have that $x \to 0$ and $y \to 0$
:when $t \to -1^+$ we have that $1 + t^3 \to 0+$, and so:
::$x \to -\infty$
::$y \to +\infty$
:when $t \to -1^-$ ... | Consider the [[Definition:Folium of Descartes|folium of Descartes]] $F$, given in [[Definition:Folium of Descartes/Parametric Form|parametric form]] as:
:$\begin {cases} x = \dfrac {3 a t} {1 + t^3} \\ y = \dfrac {3 a t^2} {1 + t^3} \end {cases}$
The [[Definition:Straight Line|straight line]] whose [[Equation of Stra... | First we note that from [[Behaviour of Parametric Equations for Folium of Descartes according to Parameter]]:
:when $t = 0$ we have that $x = y = 0$
:when $t \to \pm \infty$ we have that $x \to 0$ and $y \to 0$
:when $t \to -1^+$ we have that $1 + t^3 \to 0+$, and so:
::$x \to -\infty$
::$y \to +\infty$
:when $t \t... | Asymptote to Folium of Descartes | https://proofwiki.org/wiki/Asymptote_to_Folium_of_Descartes | https://proofwiki.org/wiki/Asymptote_to_Folium_of_Descartes | [
"Folium of Descartes"
] | [
"Definition:Folium of Descartes",
"Definition:Folium of Descartes/Parametric Form",
"Definition:Line/Straight Line",
"Equation of Straight Line in Plane/General Equation",
"Definition:Asymptote"
] | [
"Behaviour of Parametric Equations for Folium of Descartes according to Parameter",
"Sum of Two Odd Powers/Examples/Sum of Two Cubes"
] |
proofwiki-16626 | Behaviour of Parametric Equations for Folium of Descartes according to Parameter | Consider the folium of Descartes $F$, given in parametric form as:
:$\begin {cases} x = \dfrac {3 a t} {1 + t^3} \\ y = \dfrac {3 a t^2} {1 + t^3} \end {cases}$
Then:
:$F$ has a discontinuity at $t = -1$.
:For $t < -1$, the section in the $4$th quadrant is generated
:For $-1 < t \le 0$, the section in the $2$nd quadran... | :500px | Consider the [[Definition:Folium of Descartes|folium of Descartes]] $F$, given in [[Definition:Folium of Descartes/Parametric Form|parametric form]] as:
:$\begin {cases} x = \dfrac {3 a t} {1 + t^3} \\ y = \dfrac {3 a t^2} {1 + t^3} \end {cases}$
Then:
:$F$ has a discontinuity at $t = -1$.
:For $t < -1$, the section... | :[[File:FoliumOfDescartes.png|500px]] | Behaviour of Parametric Equations for Folium of Descartes according to Parameter | https://proofwiki.org/wiki/Behaviour_of_Parametric_Equations_for_Folium_of_Descartes_according_to_Parameter | https://proofwiki.org/wiki/Behaviour_of_Parametric_Equations_for_Folium_of_Descartes_according_to_Parameter | [
"Folium of Descartes"
] | [
"Definition:Folium of Descartes",
"Definition:Folium of Descartes/Parametric Form",
"Definition:Cartesian Plane/Quadrants/Fourth",
"Definition:Cartesian Plane/Quadrants/Second",
"Definition:Cartesian Plane/Quadrants/First"
] | [
"File:FoliumOfDescartes.png"
] |
proofwiki-16627 | Parametric Equation of Involute of Circle | Let $C$ be a circle of radius $a$ whose center is at the origin of a cartesian plane.
The involute $V$ of $C$ can be described by the parametric equation:
:$\begin {cases} x = a \paren {\cos \theta + \theta \sin \theta} \\ y = a \paren {\sin \theta - \theta \cos \theta} \end {cases}$ | By definition the involute of $C$ is described by the endpoint of a string unwinding from $C$.
Let that endpoint start at $\tuple {a, 0}$ on the circumference of $C$.
:500px
Let $P = \tuple {x, y}$ be an arbitrary point on $V$.
Let $Q$ be the point at which the cord is tangent to $C$.
Then $PQ$ equals the arc of $C$ fr... | Let $C$ be a [[Definition:Circle|circle]] of [[Definition:Radius of Circle|radius]] $a$ whose [[Definition:Center of Circle|center]] is at the [[Definition:Origin|origin]] of a [[Definition:Cartesian Plane|cartesian plane]].
The [[Definition:Involute|involute]] $V$ of $C$ can be described by the [[Definition:Parametri... | By definition the [[Definition:Involute|involute]] of $C$ is described by the endpoint of a string unwinding from $C$.
Let that endpoint start at $\tuple {a, 0}$ on the [[Definition:Circumference of Circle|circumference]] of $C$.
:[[File:Involute-of-Circle.png|500px]]
Let $P = \tuple {x, y}$ be an arbitrary [[Defi... | Parametric Equation of Involute of Circle | https://proofwiki.org/wiki/Parametric_Equation_of_Involute_of_Circle | https://proofwiki.org/wiki/Parametric_Equation_of_Involute_of_Circle | [
"Involutes",
"Spirals",
"Circles"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Definition:Involute",
"Definition:Parametric Equation"
] | [
"Definition:Involute",
"Definition:Circle/Circumference",
"File:Involute-of-Circle.png",
"Definition:Point",
"Definition:Point",
"Definition:Tangent Line/Circle",
"Definition:Circle/Arc",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Cosine of Difference",
"Sine of Difference"
] |
proofwiki-16628 | Equation of Ovals of Cassini/Cartesian Form | The Cartesian equation:
:$\paren {x^2 + y^2 + a^2}^2 - 4 a^2 x^2 = b^4$
describes the '''ovals of Cassini'''. | The '''ovals of Cassini''' are the loci of points $M$ in the plane such that:
:$P_1 M \times P_2 M = b^2$
Let $b$ be chosen.
Let $P = \tuple {x, y}$ be an arbitrary point of $M$.
We have:
{{begin-eqn}}
{{eqn | l = P_1 P
| r = \sqrt {\paren {x - a}^2 + y^2}
| c = Distance Formula
}}
{{eqn | l = P_2 P
|... | The [[Definition:Cartesian Equation|Cartesian equation]]:
:$\paren {x^2 + y^2 + a^2}^2 - 4 a^2 x^2 = b^4$
describes the '''[[Definition:Ovals of Cassini|ovals of Cassini]]'''. | The '''[[Definition:Ovals of Cassini|ovals of Cassini]]''' are the [[Definition:Locus|loci]] of [[Definition:Point|points]] $M$ in [[Definition:The Plane|the plane]] such that:
:$P_1 M \times P_2 M = b^2$
Let $b$ be chosen.
Let $P = \tuple {x, y}$ be an arbitrary [[Definition:Point|point]] of $M$.
We have:
{{begin-... | Equation of Ovals of Cassini/Cartesian Form | https://proofwiki.org/wiki/Equation_of_Ovals_of_Cassini/Cartesian_Form | https://proofwiki.org/wiki/Equation_of_Ovals_of_Cassini/Cartesian_Form | [
"Ovals of Cassini"
] | [
"Definition:Cartesian Equation",
"Definition:Ovals of Cassini"
] | [
"Definition:Ovals of Cassini",
"Definition:Locus",
"Definition:Point",
"Definition:Plane Surface/The Plane",
"Definition:Point",
"Distance Formula",
"Distance Formula",
"Difference of Two Squares"
] |
proofwiki-16629 | Equation of Ovals of Cassini/Polar Form | The polar equation:
:$r^4 + a^4 - 2 a^2 r^2 \cos 2 \theta = b^4$
describes the '''ovals of Cassini'''. | The '''ovals of Cassini''' are the loci of points $M$ in the plane such that:
:$P_1 M \times P_2 M = b^2$
Let $b$ be chosen.
Let $P = \tuple {x, y}$ be an arbitrary point of $M$.
Let $d_1 = \size {P_1 P}$ and $d_2 = \size {P_2 P}$.
We have:
{{begin-eqn}}
{{eqn | l = b^2
| r = d_1 d_2
| c = {{Defof|Ovals of ... | The [[Definition:Polar Equation|polar equation]]:
:$r^4 + a^4 - 2 a^2 r^2 \cos 2 \theta = b^4$
describes the '''[[Definition:Ovals of Cassini|ovals of Cassini]]'''. | The '''[[Definition:Ovals of Cassini|ovals of Cassini]]''' are the [[Definition:Locus|loci]] of [[Definition:Point|points]] $M$ in [[Definition:The Plane|the plane]] such that:
:$P_1 M \times P_2 M = b^2$
Let $b$ be chosen.
Let $P = \tuple {x, y}$ be an arbitrary [[Definition:Point|point]] of $M$.
Let $d_1 = \size ... | Equation of Ovals of Cassini/Polar Form | https://proofwiki.org/wiki/Equation_of_Ovals_of_Cassini/Polar_Form | https://proofwiki.org/wiki/Equation_of_Ovals_of_Cassini/Polar_Form | [
"Ovals of Cassini"
] | [
"Definition:Polar Equation",
"Definition:Ovals of Cassini"
] | [
"Definition:Ovals of Cassini",
"Definition:Locus",
"Definition:Point",
"Definition:Plane Surface/The Plane",
"Definition:Point",
"Law of Cosines",
"Cosine of Supplementary Angle",
"Difference of Two Squares",
"Square of Sum"
] |
proofwiki-16630 | Lemniscate of Bernoulli is Special Case of Ovals of Cassini | The lemniscate of Bernoulli is a special case of the ovals of Cassini. | The ovals of Cassini can be defined by a Cartesian equation as follows:
{{:Equation of Ovals of Cassini/Cartesian Form}}
The lemniscate of Bernoulli can be defined by a Cartesian equation as follows:
{{:Definition:Lemniscate of Bernoulli/Cartesian Definition}}
Setting $b = a$:
{{begin-eqn}}
{{eqn | l = \paren {x^2 + y^... | The [[Definition:Lemniscate of Bernoulli|lemniscate of Bernoulli]] is a special case of the [[Definition:Ovals of Cassini|ovals of Cassini]]. | The [[Definition:Ovals of Cassini|ovals of Cassini]] can be defined by a [[Equation of Ovals of Cassini/Cartesian Form|Cartesian equation]] as follows:
{{:Equation of Ovals of Cassini/Cartesian Form}}
The [[Definition:Lemniscate of Bernoulli|lemniscate of Bernoulli]] can be defined by a [[Definition:Lemniscate of Be... | Lemniscate of Bernoulli is Special Case of Ovals of Cassini/Cartesian Proof | https://proofwiki.org/wiki/Lemniscate_of_Bernoulli_is_Special_Case_of_Ovals_of_Cassini | https://proofwiki.org/wiki/Lemniscate_of_Bernoulli_is_Special_Case_of_Ovals_of_Cassini/Cartesian_Proof | [
"Lemniscate of Bernoulli is Special Case of Ovals of Cassini",
"Lemniscate of Bernoulli",
"Ovals of Cassini"
] | [
"Definition:Lemniscate of Bernoulli",
"Definition:Ovals of Cassini"
] | [
"Definition:Ovals of Cassini",
"Equation of Ovals of Cassini/Cartesian Form",
"Definition:Lemniscate of Bernoulli",
"Definition:Lemniscate of Bernoulli/Cartesian Definition",
"Definition:Lemniscate of Bernoulli",
"Definition:Ovals of Cassini"
] |
proofwiki-16631 | Lemniscate of Bernoulli is Special Case of Ovals of Cassini | The lemniscate of Bernoulli is a special case of the ovals of Cassini. | The ovals of Cassini are defined as follows:
{{:Definition:Ovals of Cassini}}
The lemniscate of Bernoulli is defined geometrically as:
{{:Definition:Lemniscate of Bernoulli/Geometric Definition}}
It follows that the lemniscate of Bernoulli is an oval of Cassini where $b = a$.
{{qed}} | The [[Definition:Lemniscate of Bernoulli|lemniscate of Bernoulli]] is a special case of the [[Definition:Ovals of Cassini|ovals of Cassini]]. | The [[Definition:Ovals of Cassini|ovals of Cassini]] are defined as follows:
{{:Definition:Ovals of Cassini}}
The [[Definition:Lemniscate of Bernoulli|lemniscate of Bernoulli]] is defined [[Definition:Lemniscate of Bernoulli/Geometric Definition|geometrically]] as:
{{:Definition:Lemniscate of Bernoulli/Geometric De... | Lemniscate of Bernoulli is Special Case of Ovals of Cassini/Geometric Proof | https://proofwiki.org/wiki/Lemniscate_of_Bernoulli_is_Special_Case_of_Ovals_of_Cassini | https://proofwiki.org/wiki/Lemniscate_of_Bernoulli_is_Special_Case_of_Ovals_of_Cassini/Geometric_Proof | [
"Lemniscate of Bernoulli is Special Case of Ovals of Cassini",
"Lemniscate of Bernoulli",
"Ovals of Cassini"
] | [
"Definition:Lemniscate of Bernoulli",
"Definition:Ovals of Cassini"
] | [
"Definition:Ovals of Cassini",
"Definition:Lemniscate of Bernoulli",
"Definition:Lemniscate of Bernoulli/Geometric Definition",
"Definition:Lemniscate of Bernoulli",
"Definition:Ovals of Cassini"
] |
proofwiki-16632 | Irrational Number divided by Rational Number is Irrational | Let $x$ be a irrational number.
Let $y$ be a non-zero rational number.
Then:
:$\dfrac x y$
is irrational. | {{AimForCont}} $\dfrac x y$ is rational number.
Then there exists an integer $p_1$ and a natural number $q_1$ such that:
:$\dfrac x y = \dfrac {p_1} {q_1}$
That is:
:$x = \dfrac {p_1} {q_1} y$
From the fact that $y$ is rational, we similarly have that there exists an integer $p_2$ and a natural number $q_2$ such that... | Let $x$ be a [[Definition:Irrational Number|irrational number]].
Let $y$ be a non-[[Definition:Zero (Number)|zero]] [[Definition:Rational Number|rational number]].
Then:
:$\dfrac x y$
is [[Definition:Irrational Number|irrational]]. | {{AimForCont}} $\dfrac x y$ is [[Definition:Rational Number|rational number]].
Then there exists an [[Definition:Integer|integer]] $p_1$ and a [[Definition:Natural Number|natural number]] $q_1$ such that:
:$\dfrac x y = \dfrac {p_1} {q_1}$
That is:
:$x = \dfrac {p_1} {q_1} y$
From the fact that $y$ is [[Definiti... | Irrational Number divided by Rational Number is Irrational | https://proofwiki.org/wiki/Irrational_Number_divided_by_Rational_Number_is_Irrational | https://proofwiki.org/wiki/Irrational_Number_divided_by_Rational_Number_is_Irrational | [
"Rational Numbers",
"Irrational Numbers"
] | [
"Definition:Irrational Number",
"Definition:Zero (Number)",
"Definition:Rational Number",
"Definition:Irrational Number"
] | [
"Definition:Rational Number",
"Definition:Integer",
"Definition:Natural Numbers",
"Definition:Rational Number",
"Definition:Integer",
"Definition:Natural Numbers",
"Integer Multiplication is Closed",
"Definition:Integer",
"Natural Number Multiplication is Closed",
"Definition:Natural Numbers",
"... |
proofwiki-16633 | Cramer's Rule | Let $n \in \N$.
Let $b_1, b_2, \dots, b_n$ be real numbers.
Let $\mathbf b = \tuple {b_1, b_2, \dots, b_n}^\intercal$.
Let $x_1, x_2, \dots, x_n$ be real numbers.
Let $\mathbf x = \tuple {x_1, x_2, \dots, x_n}^\intercal$.
Let $A$ be a nonsingular $n \times n$ matrix whose elements are in $\R$.
For each $i \in \set {1, ... | Let $C$ be the cofactor matrix of $A$.
By definition, $C^\intercal$ is the adjugate matrix of $A$.
Therefore by Matrix Product with Adjugate Matrix:
:$A \cdot C^\intercal = \det A \cdot I_n$
Because $A$ is nonsingular, $A^{-1}$ exists.
From Matrix is Nonsingular iff Determinant has Multiplicative Inverse, $1 / \det A$... | Let $n \in \N$.
Let $b_1, b_2, \dots, b_n$ be [[Definition:Real Number|real numbers]].
Let $\mathbf b = \tuple {b_1, b_2, \dots, b_n}^\intercal$.
Let $x_1, x_2, \dots, x_n$ be [[Definition:Real Number|real numbers]].
Let $\mathbf x = \tuple {x_1, x_2, \dots, x_n}^\intercal$.
Let $A$ be a [[Definition:Nonsingular M... | Let $C$ be the [[Definition:Cofactor Matrix|cofactor matrix]] of $A$.
By definition, $C^\intercal$ is the [[Definition:Adjugate Matrix|adjugate matrix]] of $A$.
Therefore by [[Matrix Product with Adjugate Matrix]]:
:$A \cdot C^\intercal = \det A \cdot I_n$
Because $A$ is [[Definition:Nonsingular Matrix|nonsingular]]... | Cramer's Rule | https://proofwiki.org/wiki/Cramer's_Rule | https://proofwiki.org/wiki/Cramer's_Rule | [
"Linear Algebra"
] | [
"Definition:Real Number",
"Definition:Real Number",
"Definition:Nonsingular Matrix",
"Definition:Matrix",
"Definition:Matrix/Element",
"Definition:Matrix",
"Definition:Matrix/Column"
] | [
"Definition:Cofactor Matrix",
"Definition:Adjugate Matrix",
"Matrix Product with Adjugate Matrix",
"Definition:Nonsingular Matrix",
"Matrix is Nonsingular iff Determinant has Multiplicative Inverse",
"Definition:Matrix/Element",
"Definition:Matrix/Indices",
"Definition:Transpose of Matrix",
"Definit... |
proofwiki-16634 | Equation of Limaçon of Pascal/Polar Form | The limaçon of Pascal can be defined by the polar equation:
:$r = b + a \cos \theta$ | Let $C$ be a circle of diameter $a$ whose circumference passes through the origin $O$.
Let the diameter of $C$ which passes through $O$ lie on the polar axis.
Let $OQ$ be a chord of $C$.
Let $b$ be a real constant.
:400px
Let $P = \polar {r, \theta}$ denote an arbitrary point on a limaçon of Pascal $L$.
We have that:
{... | The [[Definition:Limaçon of Pascal|limaçon of Pascal]] can be defined by the [[Definition:Polar Equation|polar equation]]:
:$r = b + a \cos \theta$ | Let $C$ be a [[Definition:Circle|circle]] of [[Definition:Diameter of Circle|diameter]] $a$ whose [[Definition:Circumference of Circle|circumference]] passes through the [[Definition:Origin|origin]] $O$.
Let the [[Definition:Diameter of Circle|diameter]] of $C$ which passes through $O$ lie on the [[Definition:Polar Ax... | Equation of Limaçon of Pascal/Polar Form | https://proofwiki.org/wiki/Equation_of_Limaçon_of_Pascal/Polar_Form | https://proofwiki.org/wiki/Equation_of_Limaçon_of_Pascal/Polar_Form | [
"Limaçons of Pascal"
] | [
"Definition:Limaçon of Pascal",
"Definition:Polar Equation"
] | [
"Definition:Circle",
"Definition:Circle/Diameter",
"Definition:Circle/Circumference",
"Definition:Coordinate System/Origin",
"Definition:Circle/Diameter",
"Definition:Polar Coordinates/Polar Axis",
"Definition:Circle/Chord",
"Definition:Real Number",
"Definition:Constant",
"File:Limacon-of-Pascal.... |
proofwiki-16635 | Equation of Limaçon of Pascal/Cartesian Form | The limaçon of Pascal can be defined by the Cartesian equation:
:$\paren {x^2 + y^2 - a x}^2 = b^2 \paren {x^2 + y^2}$ | {{begin-eqn}}
{{eqn | l = r
| r = b + a \cos \theta
| c = Equation of Limaçon of Pascal: Polar Form
}}
{{eqn | ll= \leadsto
| l = r^2
| r = r b + r a \cos \theta
| c =
}}
{{eqn | ll= \leadsto
| l = x^2 + y^2
| r = b \sqrt {x^2 + y^2} + a x
| c = Conversion between Cartes... | The [[Definition:Limaçon of Pascal|limaçon of Pascal]] can be defined by the [[Definition:Cartesian Equation|Cartesian equation]]:
:$\paren {x^2 + y^2 - a x}^2 = b^2 \paren {x^2 + y^2}$ | {{begin-eqn}}
{{eqn | l = r
| r = b + a \cos \theta
| c = [[Equation of Limaçon of Pascal/Polar Form|Equation of Limaçon of Pascal: Polar Form]]
}}
{{eqn | ll= \leadsto
| l = r^2
| r = r b + r a \cos \theta
| c =
}}
{{eqn | ll= \leadsto
| l = x^2 + y^2
| r = b \sqrt {x^2 + y^2... | Equation of Limaçon of Pascal/Cartesian Form | https://proofwiki.org/wiki/Equation_of_Limaçon_of_Pascal/Cartesian_Form | https://proofwiki.org/wiki/Equation_of_Limaçon_of_Pascal/Cartesian_Form | [
"Limaçons of Pascal"
] | [
"Definition:Limaçon of Pascal",
"Definition:Cartesian Equation"
] | [
"Equation of Limaçon of Pascal/Polar Form",
"Conversion between Cartesian and Polar Coordinates in Plane",
"Definition:Square/Function"
] |
proofwiki-16636 | Equation of Cissoid of Diocles/Polar Form | The '''cissoid of Diocles''' can be defined by the polar equation:
:$r = 2 a \sin \theta \tan \theta$ | :400px
By construction:
{{begin-eqn}}
{{eqn | l = OS
| r = 2 a \sec \theta
| c = {{Defof|Secant Function}}
}}
{{eqn | l = OR
| r = 2 a \cos \theta
| c = {{Defof|Cosine}}
}}
{{eqn | l = OP
| r = RS
| c = {{Defof|Cissoid of Diocles}}
}}
{{eqn | ll= \leadsto
| l = OP
| r = O... | The '''[[Definition:Cissoid of Diocles|cissoid of Diocles]]''' can be defined by the [[Definition:Polar Equation|polar equation]]:
:$r = 2 a \sin \theta \tan \theta$ | :[[File:CissoidOfDiocles.png|400px]]
By construction:
{{begin-eqn}}
{{eqn | l = OS
| r = 2 a \sec \theta
| c = {{Defof|Secant Function}}
}}
{{eqn | l = OR
| r = 2 a \cos \theta
| c = {{Defof|Cosine}}
}}
{{eqn | l = OP
| r = RS
| c = {{Defof|Cissoid of Diocles}}
}}
{{eqn | ll= \lea... | Equation of Cissoid of Diocles/Polar Form | https://proofwiki.org/wiki/Equation_of_Cissoid_of_Diocles/Polar_Form | https://proofwiki.org/wiki/Equation_of_Cissoid_of_Diocles/Polar_Form | [
"Cissoid of Diocles"
] | [
"Definition:Cissoid of Diocles",
"Definition:Polar Equation"
] | [
"File:CissoidOfDiocles.png",
"Secant Minus Cosine"
] |
proofwiki-16637 | Equation of Cissoid of Diocles/Cartesian Form | The '''cissoid of Diocles''' can be defined by the Cartesian equation:
:$x \paren {x^2 + y^2} = 2 a y^2$ | {{begin-eqn}}
{{eqn | l = r
| r = 2 a \sin \theta \tan \theta
| c = Equation of Cissoid of Diocles: Polar Form
}}
{{eqn | ll= \leadsto
| l = r^2
| r = 2 a r \sin \theta \tan \theta
| c =
}}
{{eqn | ll= \leadsto
| l = x^2 + y^2
| r = 2 a y \dfrac y x
| c = Conversion betw... | The '''[[Definition:Cissoid of Diocles|cissoid of Diocles]]''' can be defined by the [[Definition:Cartesian Equation|Cartesian equation]]:
:$x \paren {x^2 + y^2} = 2 a y^2$ | {{begin-eqn}}
{{eqn | l = r
| r = 2 a \sin \theta \tan \theta
| c = [[Equation of Cissoid of Diocles/Polar Form|Equation of Cissoid of Diocles: Polar Form]]
}}
{{eqn | ll= \leadsto
| l = r^2
| r = 2 a r \sin \theta \tan \theta
| c =
}}
{{eqn | ll= \leadsto
| l = x^2 + y^2
| r ... | Equation of Cissoid of Diocles/Cartesian Form | https://proofwiki.org/wiki/Equation_of_Cissoid_of_Diocles/Cartesian_Form | https://proofwiki.org/wiki/Equation_of_Cissoid_of_Diocles/Cartesian_Form | [
"Cissoid of Diocles"
] | [
"Definition:Cissoid of Diocles",
"Definition:Cartesian Equation"
] | [
"Equation of Cissoid of Diocles/Polar Form",
"Conversion between Cartesian and Polar Coordinates in Plane"
] |
proofwiki-16638 | Equation of Cissoid of Diocles/Parametric Form | The cissoid of Diocles can be defined by the parametric equation:
:$\begin {cases} x = 2 a \sin^2 \theta \\ \\ y = \dfrac {2 a \sin^3 \theta} {\cos \theta} \end {cases}$ | {{begin-eqn}}
{{eqn | l = r
| r = 2 a \sin \theta \tan \theta
| c = Equation of Cissoid of Diocles: Polar Form
}}
{{eqn | ll= \leadsto
| l = r \cos \theta
| r = 2 a \cos \sin \theta \paren {\dfrac {\sin \theta} {\cos \theta} }
| c =
}}
{{eqn | l = r \sin \theta
| r = 2 a \sin \theta... | The [[Definition:Cissoid of Diocles|cissoid of Diocles]] can be defined by the [[Definition:Parametric Equation|parametric equation]]:
:$\begin {cases} x = 2 a \sin^2 \theta \\ \\ y = \dfrac {2 a \sin^3 \theta} {\cos \theta} \end {cases}$ | {{begin-eqn}}
{{eqn | l = r
| r = 2 a \sin \theta \tan \theta
| c = [[Equation of Cissoid of Diocles/Polar Form|Equation of Cissoid of Diocles: Polar Form]]
}}
{{eqn | ll= \leadsto
| l = r \cos \theta
| r = 2 a \cos \sin \theta \paren {\dfrac {\sin \theta} {\cos \theta} }
| c =
}}
{{eqn |... | Equation of Cissoid of Diocles/Parametric Form | https://proofwiki.org/wiki/Equation_of_Cissoid_of_Diocles/Parametric_Form | https://proofwiki.org/wiki/Equation_of_Cissoid_of_Diocles/Parametric_Form | [
"Cissoid of Diocles"
] | [
"Definition:Cissoid of Diocles",
"Definition:Parametric Equation"
] | [
"Equation of Cissoid of Diocles/Polar Form",
"Conversion between Cartesian and Polar Coordinates in Plane"
] |
proofwiki-16639 | Primitive of One plus x Squared over One plus Fourth Power of x | :$\ds \int \frac {x^2 + 1} {x^4 + 1} \rd x = \frac 1 {\sqrt 2} \map \arctan {\frac 1 {\sqrt 2} \paren {x - \frac 1 x} } + C$ | We have:
{{begin-eqn}}
{{eqn | l = \int \frac {x^2 + 1} {x^4 + 1} \rd x
| r = \int \frac {1 + \frac 1 {x^2} } {x^2 + \frac 1 {x^2} } \rd x
}}
{{eqn | r = \int \frac {1 + \frac 1 {x^2} } {\paren {x - \frac 1 x}^2 + 2} \rd x
| c = Completing the Square
}}
{{end-eqn}}
Note that, by Derivative of Power:
:$\dfrac \d {\d ... | :$\ds \int \frac {x^2 + 1} {x^4 + 1} \rd x = \frac 1 {\sqrt 2} \map \arctan {\frac 1 {\sqrt 2} \paren {x - \frac 1 x} } + C$ | We have:
{{begin-eqn}}
{{eqn | l = \int \frac {x^2 + 1} {x^4 + 1} \rd x
| r = \int \frac {1 + \frac 1 {x^2} } {x^2 + \frac 1 {x^2} } \rd x
}}
{{eqn | r = \int \frac {1 + \frac 1 {x^2} } {\paren {x - \frac 1 x}^2 + 2} \rd x
| c = [[Completing the Square]]
}}
{{end-eqn}}
Note that, by [[Derivative of Power]]:
:$\df... | Primitive of One plus x Squared over One plus Fourth Power of x | https://proofwiki.org/wiki/Primitive_of_One_plus_x_Squared_over_One_plus_Fourth_Power_of_x | https://proofwiki.org/wiki/Primitive_of_One_plus_x_Squared_over_One_plus_Fourth_Power_of_x | [
"Primitives involving Reciprocals"
] | [] | [
"Completing the Square",
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form",
"Category:Primitives involving Reciprocals"
] |
proofwiki-16640 | Primitive of Minus One plus x Squared over One plus Fourth Power of x | :$\ds \int \frac {x^2 - 1} {x^4 + 1} \rd x = \frac 1 {2 \sqrt 2} \ln \size {\frac {x^2 - \sqrt 2 x + 1} {x^2 + \sqrt 2 x + 1} } + C$ | We have:
{{begin-eqn}}
{{eqn | l = \int \frac {x^2 - 1} {x^4 + 1} \rd x
| r = \int \frac {1 - \frac 1 {x^2} } {x^2 + \frac 1 {x^2} } \rd x
}}
{{eqn | r = \int \frac {1 - \frac 1 {x^2} } {\paren {x + \frac 1 x}^2 - 2} \rd x
| c = Completing the Square
}}
{{end-eqn}}
Note that, by Derivative of Power:
:$\dfrac \d {\d ... | :$\ds \int \frac {x^2 - 1} {x^4 + 1} \rd x = \frac 1 {2 \sqrt 2} \ln \size {\frac {x^2 - \sqrt 2 x + 1} {x^2 + \sqrt 2 x + 1} } + C$ | We have:
{{begin-eqn}}
{{eqn | l = \int \frac {x^2 - 1} {x^4 + 1} \rd x
| r = \int \frac {1 - \frac 1 {x^2} } {x^2 + \frac 1 {x^2} } \rd x
}}
{{eqn | r = \int \frac {1 - \frac 1 {x^2} } {\paren {x + \frac 1 x}^2 - 2} \rd x
| c = [[Completing the Square]]
}}
{{end-eqn}}
Note that, by [[Derivative of Power]]:
:$\df... | Primitive of Minus One plus x Squared over One plus Fourth Power of x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Minus_One_plus_x_Squared_over_One_plus_Fourth_Power_of_x | https://proofwiki.org/wiki/Primitive_of_Minus_One_plus_x_Squared_over_One_plus_Fourth_Power_of_x/Proof_1 | [
"Primitives involving Reciprocals",
"Primitives involving x to the fourth plus or minus a to the fourth",
"Primitive of Minus One plus x Squared over One plus Fourth Power of x"
] | [] | [
"Completing the Square",
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form"
] |
proofwiki-16641 | Primitive of Minus One plus x Squared over One plus Fourth Power of x | :$\ds \int \frac {x^2 - 1} {x^4 + 1} \rd x = \frac 1 {2 \sqrt 2} \ln \size {\frac {x^2 - \sqrt 2 x + 1} {x^2 + \sqrt 2 x + 1} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^2 - 1} {x^4 + 1} \rd x
| r = \int \frac {x^2} {x^4 + 1} \rd x - \int \frac 1 {x^4 + 1} \rd x
| c = Linear Combination of Primitives
}}
{{eqn | r = \frac 1 {4 \sqrt 2} \map \ln {\frac {x^2 - x \sqrt 2 + 1} {x^2 + x \sqrt 2 + 1} } - \frac 1 {2 \sqrt 2} \paren {\map \arc... | :$\ds \int \frac {x^2 - 1} {x^4 + 1} \rd x = \frac 1 {2 \sqrt 2} \ln \size {\frac {x^2 - \sqrt 2 x + 1} {x^2 + \sqrt 2 x + 1} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^2 - 1} {x^4 + 1} \rd x
| r = \int \frac {x^2} {x^4 + 1} \rd x - \int \frac 1 {x^4 + 1} \rd x
| c = [[Linear Combination of Primitives]]
}}
{{eqn | r = \frac 1 {4 \sqrt 2} \map \ln {\frac {x^2 - x \sqrt 2 + 1} {x^2 + x \sqrt 2 + 1} } - \frac 1 {2 \sqrt 2} \paren {\map ... | Primitive of Minus One plus x Squared over One plus Fourth Power of x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Minus_One_plus_x_Squared_over_One_plus_Fourth_Power_of_x | https://proofwiki.org/wiki/Primitive_of_Minus_One_plus_x_Squared_over_One_plus_Fourth_Power_of_x/Proof_2 | [
"Primitives involving Reciprocals",
"Primitives involving x to the fourth plus or minus a to the fourth",
"Primitive of Minus One plus x Squared over One plus Fourth Power of x"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of x squared over x fourth plus a fourth",
"Primitive of Reciprocal of One plus Fourth Power of x"
] |
proofwiki-16642 | Dissection of Square into 8 Acute Triangles | A square can be dissected into $8$ acute triangles. | {{ProofWanted|Trivial but wordy and tedious.}}
Category:Squares
Category:Dissections
Category:Acute Triangles
0osvjocw3tjip717wonyq7wrcbw7kre | A [[Definition:Square (Geometry)|square]] can be [[Definition:Dissection|dissected]] into $8$ [[Definition:Acute Triangle|acute triangles]]. | {{ProofWanted|Trivial but wordy and tedious.}}
[[Category:Squares]]
[[Category:Dissections]]
[[Category:Acute Triangles]]
0osvjocw3tjip717wonyq7wrcbw7kre | Dissection of Square into 8 Acute Triangles | https://proofwiki.org/wiki/Dissection_of_Square_into_8_Acute_Triangles | https://proofwiki.org/wiki/Dissection_of_Square_into_8_Acute_Triangles | [
"Squares",
"Dissections",
"Acute Triangles"
] | [
"Definition:Quadrilateral/Square",
"Definition:Dissection",
"Definition:Triangle (Geometry)/Acute"
] | [
"Category:Squares",
"Category:Dissections",
"Category:Acute Triangles"
] |
proofwiki-16643 | Dissection of Square into 9 Acute Triangles | A square can be dissected into $9$ acute triangles. | {{ProofWanted|Trivial but wordy and tedious.}} | A [[Definition:Square (Geometry)|square]] can be [[Definition:Dissection|dissected]] into $9$ [[Definition:Acute Triangle|acute triangles]]. | {{ProofWanted|Trivial but wordy and tedious.}} | Dissection of Square into 9 Acute Triangles | https://proofwiki.org/wiki/Dissection_of_Square_into_9_Acute_Triangles | https://proofwiki.org/wiki/Dissection_of_Square_into_9_Acute_Triangles | [
"Squares",
"Dissections",
"Acute Triangles"
] | [
"Definition:Quadrilateral/Square",
"Definition:Dissection",
"Definition:Triangle (Geometry)/Acute"
] | [] |
proofwiki-16644 | Length of Chord Projected from Point on Intersecting Circle | Let $C_1$ and $C_2$ be two circles which intersect at $A$ and $B$.
Let $T$ be a point on $C_1$.
Let $P$ and $Q$ be the points $TA$ and $TB$ intersect $C_2$.
:400px
Then $PQ$ is constant, wherever $T$ is positioned on $C_1$. | {{ProofWanted}}
Outline of proof:
By angle in the same segment, $\angle APB$ and $\angle ATB$ are constant.
Hence $\angle PBQ$ is constant.
This forces $PQ$ to be constant.
However this must be divided into multiple cases as sometimes $P, Q, T$ are contained in $C_1$.
The limit cases $T = A$ and $T = B$ can also be pro... | Let $C_1$ and $C_2$ be two [[Definition:Circle|circles]] which [[Definition:Intersection (Geometry)|intersect]] at $A$ and $B$.
Let $T$ be a [[Definition:Point|point]] on $C_1$.
Let $P$ and $Q$ be the [[Definition:Point|points]] $TA$ and $TB$ [[Definition:Intersection (Geometry)|intersect]] $C_2$.
:[[File:Chord-Pro... | {{ProofWanted}}
Outline of proof:
By angle in the same segment, $\angle APB$ and $\angle ATB$ are constant.
Hence $\angle PBQ$ is constant.
This forces $PQ$ to be constant.
However this must be divided into multiple cases as sometimes $P, Q, T$ are contained in $C_1$.
The limit cases $T = A$ and $T = B$ can also ... | Length of Chord Projected from Point on Intersecting Circle | https://proofwiki.org/wiki/Length_of_Chord_Projected_from_Point_on_Intersecting_Circle | https://proofwiki.org/wiki/Length_of_Chord_Projected_from_Point_on_Intersecting_Circle | [
"Circles"
] | [
"Definition:Circle",
"Definition:Intersection (Geometry)",
"Definition:Point",
"Definition:Point",
"Definition:Intersection (Geometry)",
"File:Chord-Projected-from-Point-on-Intersecting-Circle.png",
"Definition:Constant"
] | [] |
proofwiki-16645 | Regiomontanus' Angle Maximization Problem | Let $AB$ be a line segment.
Let $AB$ be produced to $P$.
Let $PQ$ be constructed perpendicular to $AB$.
Then the angle $AQB$ is greatest when $PQ$ is tangent to a circle passing through $A$, $B$ and $Q$. | :400px
There exists a unique circle $C$ tangent to $PQ$ passing through $A$ and $B$.
From Angles in Same Segment of Circle are Equal, the angle subtended by $AB$ from any point on $C$ is the same as angle $\angle AQB$.
All other points on $PQ$ that are not $Q$ itself are outside $C$.
Hence the angle subtended by $AB$ t... | Let $AB$ be a [[Definition:Line Segment|line segment]].
Let $AB$ be [[Definition:Production|produced]] to $P$.
Let $PQ$ be constructed [[Definition:Perpendicular|perpendicular]] to $AB$.
Then the [[Definition:Angle|angle]] $AQB$ is greatest when $PQ$ is [[Definition:Tangent to Circle|tangent]] to a [[Definition:Cir... | :[[File:Regiomontanus-angle-min-problem.png|400px]]
There exists a [[Definition:Unique|unique]] [[Definition:Circle|circle]] $C$ [[Definition:Tangent to Circle|tangent]] to $PQ$ passing through $A$ and $B$.
From [[Angles in Same Segment of Circle are Equal]], the [[Definition:Angle|angle]] [[Definition:Subtend|subten... | Regiomontanus' Angle Maximization Problem | https://proofwiki.org/wiki/Regiomontanus'_Angle_Maximization_Problem | https://proofwiki.org/wiki/Regiomontanus'_Angle_Maximization_Problem | [
"Circles",
"Classic Problems"
] | [
"Definition:Line/Segment",
"Definition:Production",
"Definition:Right Angle/Perpendicular",
"Definition:Angle",
"Definition:Tangent Line/Circle",
"Definition:Circle"
] | [
"File:Regiomontanus-angle-min-problem.png",
"Definition:Unique",
"Definition:Circle",
"Definition:Tangent Line/Circle",
"Angles in Same Segment of Circle are Equal",
"Definition:Angle",
"Definition:Subtend",
"Definition:Point",
"Definition:Angle",
"Definition:Point",
"Definition:Angle",
"Defin... |
proofwiki-16646 | First Projection on Ordered Pair of Sets | Let $a$ and $b$ be sets.
Let $w = \tuple {a, b}$ denote the ordered pair of $a$ and $b$.
Let $\map {\pr_1} w$ denote the first projection on $w$.
Then:
:$\ds \map {\pr_1} w = \bigcup \bigcap w$
where $\ds \bigcup$ and $\ds \bigcap$ denote union and intersection respectively. | We have by definition of first projection that:
:$\map {\pr_1} w = \map {\pr_1} {a, b} = a$
Then:
{{begin-eqn}}
{{eqn | l = \bigcup \bigcap w
| r = \bigcup \bigcap \tuple {a, b}
| c = Definition of $w$
}}
{{eqn | r = \bigcup \bigcap \set {\set a, \set {a, b} }
| c = {{Defof|Kuratowski Formalization of... | Let $a$ and $b$ be [[Definition:Set|sets]].
Let $w = \tuple {a, b}$ denote the [[Definition:Ordered Pair|ordered pair]] of $a$ and $b$.
Let $\map {\pr_1} w$ denote the [[Definition:First Projection|first projection]] on $w$.
Then:
:$\ds \map {\pr_1} w = \bigcup \bigcap w$
where $\ds \bigcup$ and $\ds \bigcap$ deno... | We have by [[Definition:First Projection|definition of first projection]] that:
:$\map {\pr_1} w = \map {\pr_1} {a, b} = a$
Then:
{{begin-eqn}}
{{eqn | l = \bigcup \bigcap w
| r = \bigcup \bigcap \tuple {a, b}
| c = Definition of $w$
}}
{{eqn | r = \bigcup \bigcap \set {\set a, \set {a, b} }
| c = ... | First Projection on Ordered Pair of Sets | https://proofwiki.org/wiki/First_Projection_on_Ordered_Pair_of_Sets | https://proofwiki.org/wiki/First_Projection_on_Ordered_Pair_of_Sets | [
"Ordered Pairs",
"Projections"
] | [
"Definition:Set",
"Definition:Ordered Pair",
"Definition:Projection (Mapping Theory)/First Projection",
"Definition:Set Union/Set of Sets",
"Definition:Set Intersection/Set of Sets"
] | [
"Definition:Projection (Mapping Theory)/First Projection",
"Intersection of Doubleton",
"Union of Singleton"
] |
proofwiki-16647 | Intersection of Doubleton | Let $x$ and $y$ be sets.
Let $\set {x, y}$ be a doubleton.
Then $\bigcap \set {x, y}$ is a set such that:
:$\bigcap \set {x, y} = x \cap y$ | {{begin-eqn}}
{{eqn | o =
| r = z \in \bigcap \set {x, y}
| c =
}}
{{eqn | ll= \leadstoandfrom
| o =
| r = \forall w \in \set {x, y}: z \in w
| c = {{Defof|Intersection of Class}}
}}
{{eqn | ll= \leadstoandfrom
| o =
| r = \forall w: \paren {\paren {w = x \land w = y} \land... | Let $x$ and $y$ be [[Definition:Set|sets]].
Let $\set {x, y}$ be a [[Definition:Doubleton Class|doubleton]].
Then $\bigcap \set {x, y}$ is a [[Definition:Set|set]] such that:
:$\bigcap \set {x, y} = x \cap y$ | {{begin-eqn}}
{{eqn | o =
| r = z \in \bigcap \set {x, y}
| c =
}}
{{eqn | ll= \leadstoandfrom
| o =
| r = \forall w \in \set {x, y}: z \in w
| c = {{Defof|Intersection of Class}}
}}
{{eqn | ll= \leadstoandfrom
| o =
| r = \forall w: \paren {\paren {w = x \land w = y} \land... | Intersection of Doubleton | https://proofwiki.org/wiki/Intersection_of_Doubleton | https://proofwiki.org/wiki/Intersection_of_Doubleton | [
"Class Intersection",
"Doubleton Classes"
] | [
"Definition:Set",
"Definition:Doubleton/Class Theory",
"Definition:Set"
] | [
"Equality implies Substitution",
"Intersection of Non-Empty Class is Set",
"Definition:Set"
] |
proofwiki-16648 | Second Projection on Ordered Pair of Sets | Let $a$ and $b$ be sets.
Let $w = \tuple {a, b}$ denote the ordered pair of $a$ and $b$.
Let $\map {\pr_2} w$ denote the second projection on $w$.
Then:
:$\ds \map {\pr_2} w = \begin {cases} \ds \map \bigcup {\bigcup w \setminus \bigcap w} & : \ds \bigcup w \ne \bigcap w \\ \ds \bigcup \bigcup w & : \bigcup w = \ds \bi... | We have by definition of second projection that:
:$\map {\pr_1} w = \map {\pr_1} {a, b} = b$
We consider:
{{begin-eqn}}
{{eqn | l = \bigcup w
| r = \bigcup \tuple {a, b}
| c = Definition of $w$
}}
{{eqn | r = \bigcup \set {\set a, \set {a, b} }
| c = {{Defof|Kuratowski Formalization of Ordered Pair|Or... | Let $a$ and $b$ be [[Definition:Set|sets]].
Let $w = \tuple {a, b}$ denote the [[Definition:Ordered Pair|ordered pair]] of $a$ and $b$.
Let $\map {\pr_2} w$ denote the [[Definition:Second Projection|second projection]] on $w$.
Then:
:$\ds \map {\pr_2} w = \begin {cases} \ds \map \bigcup {\bigcup w \setminus \bigcap... | We have by [[Definition:Second Projection|definition of second projection]] that:
:$\map {\pr_1} w = \map {\pr_1} {a, b} = b$
We consider:
{{begin-eqn}}
{{eqn | l = \bigcup w
| r = \bigcup \tuple {a, b}
| c = Definition of $w$
}}
{{eqn | r = \bigcup \set {\set a, \set {a, b} }
| c = {{Defof|Kuratow... | Second Projection on Ordered Pair of Sets | https://proofwiki.org/wiki/Second_Projection_on_Ordered_Pair_of_Sets | https://proofwiki.org/wiki/Second_Projection_on_Ordered_Pair_of_Sets | [
"Ordered Pairs",
"Projections"
] | [
"Definition:Set",
"Definition:Ordered Pair",
"Definition:Projection (Mapping Theory)/Second Projection",
"Definition:Set Union/Set of Sets",
"Definition:Set Intersection/Set of Sets",
"Definition:Set Difference"
] | [
"Definition:Projection (Mapping Theory)/Second Projection",
"Union of Doubleton",
"Intersection of Doubleton",
"Union of Singleton",
"Union of Doubleton",
"Set Union is Idempotent",
"Union of Singleton"
] |
proofwiki-16649 | Order Isomorphism between Tosets is not necessarily Unique | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be tosets.
Let $\struct {S_1, \preccurlyeq_1} \cong \struct {S_2, \preccurlyeq_2}$, that is, let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be order isomorphic.
Then it is not necessarily the case that there is exactly one ... | Proof by Counterexample:
Let $\Z$ denote the set of integers.
We have that Integers under Usual Ordering form Totally Ordered Set.
Let $m \in \Z$ be an arbitrary integer.
Let $f_m: \Z \to \Z$ be the order isomorphism defined as:
:$\forall a \in \Z: \map {f_m} a = a + m$
It follows that there are at least as many order ... | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be [[Definition:Toset|tosets]].
Let $\struct {S_1, \preccurlyeq_1} \cong \struct {S_2, \preccurlyeq_2}$, that is, let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be [[Definition:Order Isomorphism|order isomorphic]].
Then ... | [[Proof by Counterexample]]:
Let $\Z$ denote the [[Definition:Integer|set of integers]].
We have that [[Integers under Usual Ordering form Totally Ordered Set]].
Let $m \in \Z$ be an arbitrary [[Definition:Integer|integer]].
Let $f_m: \Z \to \Z$ be the [[Definition:Order Isomorphism|order isomorphism]] defined as:... | Order Isomorphism between Tosets is not necessarily Unique | https://proofwiki.org/wiki/Order_Isomorphism_between_Tosets_is_not_necessarily_Unique | https://proofwiki.org/wiki/Order_Isomorphism_between_Tosets_is_not_necessarily_Unique | [
"Order Isomorphisms",
"Total Orderings"
] | [
"Symbols:Abbreviations/T/Toset",
"Definition:Order Isomorphism",
"Definition:Mapping",
"Definition:Order Isomorphism"
] | [
"Proof by Counterexample",
"Definition:Integer",
"Integers under Usual Ordering form Totally Ordered Set",
"Definition:Integer",
"Definition:Order Isomorphism",
"Definition:Order Isomorphism",
"Definition:Integer"
] |
proofwiki-16650 | Pi as Sum of Alternating Sequence of Products of 3 Consecutive Reciprocals/Lemma | :$\ds \iiint \dfrac x {x^2 + 1} \rd x \rd x \rd x = x \map \arctan x + \dfrac {\paren {x^2 - 1} \map \ln {x^2 + 1} - 3 x^2} 4$
with all integration constants at $0$. | First primitive:
{{begin-eqn}}
{{eqn | l = \int \dfrac x {x^2 + 1} \rd x
| r = \dfrac 1 2 \int \dfrac {2 x} {x^2 + 1} \rd x
| c =
}}
{{eqn | r = \dfrac {\map \ln {x^2 + 1} } 2
| c = Primitive of Function under its Derivative
}}
{{end-eqn}}
The integration constant is not added due to the series never... | :$\ds \iiint \dfrac x {x^2 + 1} \rd x \rd x \rd x = x \map \arctan x + \dfrac {\paren {x^2 - 1} \map \ln {x^2 + 1} - 3 x^2} 4$
with all [[Definition:Arbitrary Constant (Calculus)|integration constants]] at $0$. | First [[Definition:Primitive (Calculus)|primitive]]:
{{begin-eqn}}
{{eqn | l = \int \dfrac x {x^2 + 1} \rd x
| r = \dfrac 1 2 \int \dfrac {2 x} {x^2 + 1} \rd x
| c =
}}
{{eqn | r = \dfrac {\map \ln {x^2 + 1} } 2
| c = [[Primitive of Function under its Derivative]]
}}
{{end-eqn}}
The [[Definition:Ar... | Pi as Sum of Alternating Sequence of Products of 3 Consecutive Reciprocals/Lemma | https://proofwiki.org/wiki/Pi_as_Sum_of_Alternating_Sequence_of_Products_of_3_Consecutive_Reciprocals/Lemma | https://proofwiki.org/wiki/Pi_as_Sum_of_Alternating_Sequence_of_Products_of_3_Consecutive_Reciprocals/Lemma | [
"Pi as Sum of Alternating Sequence of Products of 3 Consecutive Reciprocals"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Definition:Primitive (Calculus)",
"Primitive of Function under its Derivative",
"Definition:Primitive (Calculus)/Constant of Integration",
"Definition:Primitive (Calculus)",
"Primitive of Logarithm of x squared plus a squared",
"Definition:Primitive (Calculus)",
"Integral of Power",
"Primitive of Arc... |
proofwiki-16651 | Supremum of Set Equals Maximum of Suprema of Subsets | Let $S$ be a non-empty real set.
Let $\set {S_i: i \in \set {1, 2, \ldots, n} }$, $n \in \N_{>0}$, be a set of non-empty subsets of $S$.
Let $S = \bigcup S_i$.
Then:
:$S_i$ has a supremum for every $i$ in $\set {1, 2, \ldots, n}$
{{iff}}:
:$S$ has a supremum
and, in either case:
:$\sup S = \max \set {\sup S_1, \sup S_2... | === Necessary Condition ===
Let:
:$S$ have a supremum.
We need to show that:
:$(1): \quad S_i$ has a supremum for every $i$ in $\set {1, 2, \ldots, n}$
:$(2): \quad \sup S = \max \set {\sup S_1, \sup S_2, \ldots, \sup S_n}$
By Supremum of Set of Real Numbers is at least Supremum of Subset, $\sup S_i$ exists for every $... | Let $S$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Real Number|real set]].
Let $\set {S_i: i \in \set {1, 2, \ldots, n} }$, $n \in \N_{>0}$, be a set of [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subsets]] of $S$.
Let $S = \bigcup S_i$.
Then:
:$S_i$ has a [[Definition:Supremum of Subset... | === Necessary Condition ===
Let:
:$S$ have a [[Definition:Supremum of Subset of Real Numbers|supremum]].
We need to show that:
:$(1): \quad S_i$ has a [[Definition:Supremum of Subset of Real Numbers|supremum]] for every $i$ in $\set {1, 2, \ldots, n}$
:$(2): \quad \sup S = \max \set {\sup S_1, \sup S_2, \ldots, \sup ... | Supremum of Set Equals Maximum of Suprema of Subsets | https://proofwiki.org/wiki/Supremum_of_Set_Equals_Maximum_of_Suprema_of_Subsets | https://proofwiki.org/wiki/Supremum_of_Set_Equals_Maximum_of_Suprema_of_Subsets | [
"Real Analysis"
] | [
"Definition:Non-Empty Set",
"Definition:Real Number",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Supremum of Set/Real Numbers",
"Definition:Supremum of Set/Real Numbers"
] | [
"Definition:Supremum of Set/Real Numbers",
"Definition:Supremum of Set/Real Numbers",
"Supremum of Set of Real Numbers is at least Supremum of Subset",
"Supremum of Set of Real Numbers is at least Supremum of Subset",
"Supremum of Subset of Union Equals Supremum of Union",
"Definition:Supremum of Set/Real... |
proofwiki-16652 | P-adic Integer has Unique Coherent Sequence Representative/Lemma 1 | :$\forall j \in \N: \norm {\beta_{\map N {j + 1} } - \beta_{\map N j} }_p \le p^{-\paren {j + 1} }$ | Let $j \in N$
Suppose $\map N {j + 1} \ge \map N j$
By definition:
:$\norm{\beta_{\map N {j + 1} } - \beta_{\map N j} }_p \le p^{-\paren {j + 1} }$
Now suppose $\map N j \ge \map N {j + 1}$
Then:
{{begin-eqn}}
{{eqn | l = \norm {\beta_{\map N {j + 1} } - \beta_{\map N j} }_p
| o = \le
| r = p^{-\paren {j +... | :$\forall j \in \N: \norm {\beta_{\map N {j + 1} } - \beta_{\map N j} }_p \le p^{-\paren {j + 1} }$ | Let $j \in N$
Suppose $\map N {j + 1} \ge \map N j$
By definition:
:$\norm{\beta_{\map N {j + 1} } - \beta_{\map N j} }_p \le p^{-\paren {j + 1} }$
Now suppose $\map N j \ge \map N {j + 1}$
Then:
{{begin-eqn}}
{{eqn | l = \norm {\beta_{\map N {j + 1} } - \beta_{\map N j} }_p
| o = \le
| r = p^{-\paren ... | P-adic Integer has Unique Coherent Sequence Representative/Lemma 1 | https://proofwiki.org/wiki/P-adic_Integer_has_Unique_Coherent_Sequence_Representative/Lemma_1 | https://proofwiki.org/wiki/P-adic_Integer_has_Unique_Coherent_Sequence_Representative/Lemma_1 | [
"P-adic Integer has Unique Coherent Sequence Representative"
] | [] | [
"Power Function on Base between Zero and One is Strictly Decreasing/Integer",
"Category:P-adic Integer has Unique Coherent Sequence Representative"
] |
proofwiki-16653 | P-adic Integer has Unique Coherent Sequence Representative/Lemma 2 | :$\forall j \in \N: \norm {\alpha_{j + 1} - \alpha_j }_p \le p^{-\paren {j + 1} }$ | For all $j \in \N$:
{{begin-eqn}}
{{eqn | l = \norm {\alpha_{j + 1} - \alpha_j}_p
| r = \norm {\alpha_{j + 1} - \gamma_{j + 1} + \gamma_{j + 1} - \gamma_j + \gamma_j - \alpha_j}_p
}}
{{eqn | o = \le
| r = \max \set {\norm {\alpha_{j + 1} - \gamma_{j + 1} }_p \mathop , \norm {\gamma_{j + 1} - \gamma_j }_p \m... | :$\forall j \in \N: \norm {\alpha_{j + 1} - \alpha_j }_p \le p^{-\paren {j + 1} }$ | For all $j \in \N$:
{{begin-eqn}}
{{eqn | l = \norm {\alpha_{j + 1} - \alpha_j}_p
| r = \norm {\alpha_{j + 1} - \gamma_{j + 1} + \gamma_{j + 1} - \gamma_j + \gamma_j - \alpha_j}_p
}}
{{eqn | o = \le
| r = \max \set {\norm {\alpha_{j + 1} - \gamma_{j + 1} }_p \mathop , \norm {\gamma_{j + 1} - \gamma_j }_p \m... | P-adic Integer has Unique Coherent Sequence Representative/Lemma 2 | https://proofwiki.org/wiki/P-adic_Integer_has_Unique_Coherent_Sequence_Representative/Lemma_2 | https://proofwiki.org/wiki/P-adic_Integer_has_Unique_Coherent_Sequence_Representative/Lemma_2 | [
"P-adic Integer has Unique Coherent Sequence Representative"
] | [] | [
"Power Function on Base between Zero and One is Strictly Decreasing/Integer",
"Category:P-adic Integer has Unique Coherent Sequence Representative"
] |
proofwiki-16654 | Differential Equation defining Confocal Conics | Consider the equation:
:$(1): \quad \dfrac {x^2} {a^2 + \lambda} + \dfrac {y^2} {b^2 + \lambda} = 1$
where $a^2 > b^2$ and $-\lambda < a^2$.
defining the set of confocal conics whose foci are at $\tuple {\pm \sqrt {a^2 - b^2}, 0}$.
The differential equation defining these confocal conics is:
:$x y \paren {\paren {y'}^2... | {{begin-eqn}}
{{eqn | l = \dfrac {x^2} {a^2 + \lambda}
| r = 1 - \dfrac {y^2} {b^2 + \lambda}
| c = from $(1)$
}}
{{eqn | r = \dfrac {b^2 + \lambda - y^2} {b^2 + \lambda}
| c =
}}
{{eqn | n = 2
| ll= \leadsto
| l = a^2 + \lambda
| r = \dfrac {x^2 \paren {b^2 + \lambda} } {b^2 + \lam... | Consider the [[Equation of Confocal Conics/Formulation 1|equation]]:
:$(1): \quad \dfrac {x^2} {a^2 + \lambda} + \dfrac {y^2} {b^2 + \lambda} = 1$
where $a^2 > b^2$ and $-\lambda < a^2$.
defining the set of [[Definition:Confocal Conics|confocal conics]] whose [[Definition:Focus of Conic Section|foci]] are at $\tuple {... | {{begin-eqn}}
{{eqn | l = \dfrac {x^2} {a^2 + \lambda}
| r = 1 - \dfrac {y^2} {b^2 + \lambda}
| c = from $(1)$
}}
{{eqn | r = \dfrac {b^2 + \lambda - y^2} {b^2 + \lambda}
| c =
}}
{{eqn | n = 2
| ll= \leadsto
| l = a^2 + \lambda
| r = \dfrac {x^2 \paren {b^2 + \lambda} } {b^2 + \lam... | Differential Equation defining Confocal Conics | https://proofwiki.org/wiki/Differential_Equation_defining_Confocal_Conics | https://proofwiki.org/wiki/Differential_Equation_defining_Confocal_Conics | [
"Confocal Conics"
] | [
"Equation of Confocal Conics/Formulation 1",
"Definition:Confocal Conics",
"Definition:Conic Section/Focus",
"Definition:Differential Equation",
"Definition:Confocal Conics"
] | [
"Definition:Differentiation"
] |
proofwiki-16655 | P-adic Integer has Unique Coherent Sequence Representative/Lemma 3 | :$\sequence {\alpha_n}$ and $\sequence {\beta_n}$ are representatives of the same $p$-adic number in $\Q_p$. | From Representatives of same P-adic Number iff Difference is Null Sequence, it needs only to be shown that $\sequence {\alpha_j - \beta_j}$ is a null sequence.
Let $\epsilon \in \R_{> 0}$.
From Sequence of Powers of Number less than One, the sequence $\sequence {p^n}$ is a null sequence.
Then there exists $j \in \N$ su... | :$\sequence {\alpha_n}$ and $\sequence {\beta_n}$ are [[Definition:Representative of P-adic Number|representatives]] of the same [[Definition:P-adic Number|$p$-adic number]] in $\Q_p$. | From [[Representatives of same P-adic Number iff Difference is Null Sequence]], it needs only to be shown that $\sequence {\alpha_j - \beta_j}$ is a [[Definition:Null Sequence in Normed Division Ring|null sequence]].
Let $\epsilon \in \R_{> 0}$.
From [[Sequence of Powers of Number less than One]], the [[Definition:S... | P-adic Integer has Unique Coherent Sequence Representative/Lemma 3 | https://proofwiki.org/wiki/P-adic_Integer_has_Unique_Coherent_Sequence_Representative/Lemma_3 | https://proofwiki.org/wiki/P-adic_Integer_has_Unique_Coherent_Sequence_Representative/Lemma_3 | [
"P-adic Integer has Unique Coherent Sequence Representative"
] | [
"Definition:P-adic Number/Representative",
"Definition:P-adic Number"
] | [
"Representatives of same P-adic Number iff Difference is Null Sequence",
"Definition:Null Sequence/Normed Division Ring",
"Sequence of Powers of Number less than One",
"Definition:Sequence",
"Definition:Null Sequence/Real Numbers",
"Definition:Null Sequence/Normed Division Ring",
"Category:P-adic Intege... |
proofwiki-16656 | P-adic Integer has Unique Coherent Sequence Representative/Lemma 4 | :$\sequence {\alpha_j}$ is the only coherent sequence that represents $a$. | Let $\sequence {\alpha'_j}$ be a coherent sequence not equal to $\sequence {\alpha_j}$.
From Representatives of same P-adic Number iff Difference is Null Sequence, it needs only to be shown that $\sequence {\alpha_j - \alpha'_j}$ is not a null sequence.
Since $\sequence {\alpha'_j} \ne \sequence{\alpha_j}$ then:
:$\exi... | :$\sequence {\alpha_j}$ is the only [[Definition:P-adically Coherent Sequence|coherent sequence]] that [[Definition:Representative of P-adic Number|represents]] $a$. | Let $\sequence {\alpha'_j}$ be a [[Definition:P-adically Coherent Sequence|coherent sequence]] not [[Definition:Equal|equal]] to $\sequence {\alpha_j}$.
From [[Representatives of same P-adic Number iff Difference is Null Sequence]], it needs only to be shown that $\sequence {\alpha_j - \alpha'_j}$ is not a [[Definiti... | P-adic Integer has Unique Coherent Sequence Representative/Lemma 4 | https://proofwiki.org/wiki/P-adic_Integer_has_Unique_Coherent_Sequence_Representative/Lemma_4 | https://proofwiki.org/wiki/P-adic_Integer_has_Unique_Coherent_Sequence_Representative/Lemma_4 | [
"P-adic Integer has Unique Coherent Sequence Representative"
] | [
"Definition:P-adically Coherent Sequence",
"Definition:P-adic Number/Representative"
] | [
"Definition:P-adically Coherent Sequence",
"Definition:Equals",
"Representatives of same P-adic Number iff Difference is Null Sequence",
"Definition:Null Sequence/Normed Division Ring",
"Definition:P-adically Coherent Sequence",
"Reduced Residue System Modulo Prime",
"Definition:P-adically Coherent Sequ... |
proofwiki-16657 | Elimination of Constants by Partial Differentiation | Let $x_1, x_2, \dotsc, x_m$ be independent variables.
Let $c_1, c_2, \dotsc, c_n$ be arbitrary constants.
Let this equation:
:$(1): \quad \map f {x_1, x_2, \dotsc, x_m, z, c_1, c_2, \dotsc, c_n} = 0$
define a dependent variable $z$ via the implicit function $f$.
Then it may be possible to eliminate the constants by suc... | We differentiate $(1)$ partially {{WRT|Differentiation}} each of $x_j$ for $1 \le j \le m$:
:$(2): \quad \dfrac {\partial f} {\partial x_j} + \dfrac {\partial f} {\partial z} \cdot \dfrac {\partial z} {\partial x_j} = 0$
Suppose $m \ge n$.
Then there exist sufficient equations of the form of $2$ for the constants $c_1,... | Let $x_1, x_2, \dotsc, x_m$ be [[Definition:Independent Variable|independent variables]].
Let $c_1, c_2, \dotsc, c_n$ be arbitrary [[Definition:Constant|constants]].
Let this equation:
:$(1): \quad \map f {x_1, x_2, \dotsc, x_m, z, c_1, c_2, \dotsc, c_n} = 0$
define a [[Definition:Dependent Variable|dependent variabl... | We [[Definition:Partial Derivative|differentiate $(1)$ partially]] {{WRT|Differentiation}} each of $x_j$ for $1 \le j \le m$:
:$(2): \quad \dfrac {\partial f} {\partial x_j} + \dfrac {\partial f} {\partial z} \cdot \dfrac {\partial z} {\partial x_j} = 0$
Suppose $m \ge n$.
Then there exist sufficient equations of th... | Elimination of Constants by Partial Differentiation | https://proofwiki.org/wiki/Elimination_of_Constants_by_Partial_Differentiation | https://proofwiki.org/wiki/Elimination_of_Constants_by_Partial_Differentiation | [
"Partial Differentiation"
] | [
"Definition:Independent Variable",
"Definition:Constant",
"Definition:Dependent Variable",
"Definition:Implicit Function",
"Definition:Constant",
"Definition:Partial Derivative"
] | [
"Definition:Partial Derivative",
"Definition:Constant",
"Definition:Partial Derivative"
] |
proofwiki-16658 | Representatives of same P-adic Number iff Difference is Null Sequence | Let $p$ be a prime number.
Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rational numbers $\Q$.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\sequence{\alpha_n}$ and $\sequence{\beta_n}$ be Cauchy sequences in $\struct {Q, \norm {\,\cdot\,}_p}$.
Then:
:$\sequence {\alpha_n}$ and $\se... | Let $\CC$ be the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$.
Let $\NN$ be the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$.
Then $\Q_p$ is the quotient ring $\CC \, \big / \NN$ by definition of the $p$-adic numbers.
Hence:
:$\sequence {\alpha_n}$ and $\sequence {\beta_... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\norm {\,\cdot\,}_p$ be the [[Definition:P-adic Norm|$p$-adic norm]] on the [[Definition:Rational Number|rational numbers]] $\Q$.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\sequence... | Let $\CC$ be the [[Definition:Ring of Cauchy Sequences|commutative ring of Cauchy sequences]] over $\struct {\Q, \norm {\,\cdot\,}_p}$.
Let $\NN$ be the [[Definition:Set|set]] of [[Definition:Null Sequence|null sequences]] in $\struct {\Q, \norm {\,\cdot\,}_p}$.
Then $\Q_p$ is the [[Definition:Quotient Ring|quotient ... | Representatives of same P-adic Number iff Difference is Null Sequence | https://proofwiki.org/wiki/Representatives_of_same_P-adic_Number_iff_Difference_is_Null_Sequence | https://proofwiki.org/wiki/Representatives_of_same_P-adic_Number_iff_Difference_is_Null_Sequence | [
"P-adic Number Theory"
] | [
"Definition:Prime Number",
"Definition:P-adic Norm",
"Definition:Rational Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:P-adic Number/Representative",
"Definition:P-adic Number",
"Definition:Sequence",
"Definition:Null Sequence/N... | [
"Definition:Ring of Cauchy Sequences",
"Definition:Set",
"Definition:Null Sequence",
"Definition:Quotient Ring",
"Definition:Field of P-adic Numbers",
"Definition:P-adic Number/Representative",
"Definition:P-adic Number",
"Definition:Element",
"Definition:Coset/Left Coset",
"Element in Left Coset ... |
proofwiki-16659 | Partial Differential Equation of Planes in 3-Space | The set of planes in real Cartesian $3$-dimensional space can be described by the system of partial differential equations:
{{begin-eqn}}
{{eqn | l = \dfrac {\partial^2 z} {\partial x^2}
| r = 0
}}
{{eqn | l = \dfrac {\partial^2 z} {\partial x \partial y}
| r = 0
}}
{{eqn | l = \dfrac {\partial^2 z} {\parti... | From Equation of Plane, we have that the equation defining a general plane $P$ is:
:$\alpha_1 x + \alpha_2 y + \alpha_3 z = \gamma$
which can be written as:
:$z = a x + b y + c$
by setting:
{{begin-eqn}}
{{eqn | l = a
| r = \dfrac {-\alpha_1} {\alpha_3}
}}
{{eqn | l = b
| r = \dfrac {-\alpha_2} {\alpha_3}
}... | The set of [[Definition:Plane|planes]] in [[Definition:Real Cartesian Space|real Cartesian $3$-dimensional space]] can be described by the [[Definition:System of Differential Equations|system]] of [[Definition:Partial Differential Equation|partial differential equations]]:
{{begin-eqn}}
{{eqn | l = \dfrac {\partial^2 ... | From [[Equation of Plane]], we have that the [[Definition:Equation|equation]] defining a general [[Definition:Plane|plane]] $P$ is:
:$\alpha_1 x + \alpha_2 y + \alpha_3 z = \gamma$
which can be written as:
:$z = a x + b y + c$
by setting:
{{begin-eqn}}
{{eqn | l = a
| r = \dfrac {-\alpha_1} {\alpha_3}
}}
{{eqn |... | Partial Differential Equation of Planes in 3-Space | https://proofwiki.org/wiki/Partial_Differential_Equation_of_Planes_in_3-Space | https://proofwiki.org/wiki/Partial_Differential_Equation_of_Planes_in_3-Space | [
"Partial Differential Equations",
"Solid Analytic Geometry"
] | [
"Definition:Plane Surface",
"Definition:Cartesian Product/Cartesian Space/Real Cartesian Space",
"Definition:Differential Equation/System",
"Definition:Differential Equation/Partial"
] | [
"Equation of Plane",
"Definition:Equation",
"Definition:Plane Surface",
"Elimination of Constants by Partial Differentiation",
"Definition:Dependent Variable",
"Definition:Independent Variable",
"Definition:Constant",
"Definition:Partial Derivative",
"Definition:Equation",
"Definition:Constant",
... |
proofwiki-16660 | Inverse of Vandermonde Matrix/Eisinberg Formula | Let:
{{begin-eqn}}
{{eqn | l = \prod_{k \mathop = 1}^n \paren {x - x_k}
| r = a_n x^n + \sum_{m \mathop = 0}^{n - 1} a_m x^m
| c = Polynomial expansion in powers of $x$
}}
{{eqn | r = x^n + \sum_{m \mathop = 0}^{n - 1} \paren {-1}^{n - m} \map {e_{n - m} } {x_1, \ldots, x_n} \, x^m
| c = Viète's Formu... | === Lemma 1 ===
Given values $z_1, \ldots, z_{p + 1}$ and $1 \le m \le p$, then:
{{begin-eqn}}
{{eqn | n = 2
| l = \ds \map {e_m} {z_1, \ldots, z_p, z_{p + 1} }
| r = z_{p + 1} \map {e_{m - 1} } {z_1, \ldots, z_p} + \map {e_m} {z_1, \ldots, z_p}
| c = Recursion Property of Elementary Symmetric Functio... | Let:
{{begin-eqn}}
{{eqn | l = \prod_{k \mathop = 1}^n \paren {x - x_k}
| r = a_n x^n + \sum_{m \mathop = 0}^{n - 1} a_m x^m
| c = Polynomial expansion in powers of $x$
}}
{{eqn | r = x^n + \sum_{m \mathop = 0}^{n - 1} \paren {-1}^{n - m} \map {e_{n - m} } {x_1, \ldots, x_n} \, x^m
| c = [[Viète's Fo... | === Lemma 1 ===
Given values $z_1, \ldots, z_{p + 1}$ and $1 \le m \le p$, then:
{{begin-eqn}}
{{eqn | n = 2
| l = \ds \map {e_m} {z_1, \ldots, z_p, z_{p + 1} }
| r = z_{p + 1} \map {e_{m - 1} } {z_1, \ldots, z_p} + \map {e_m} {z_1, \ldots, z_p}
| c = [[Recursion Property of Elementary Symmetric Fun... | Inverse of Vandermonde Matrix/Eisinberg Formula | https://proofwiki.org/wiki/Inverse_of_Vandermonde_Matrix/Eisinberg_Formula | https://proofwiki.org/wiki/Inverse_of_Vandermonde_Matrix/Eisinberg_Formula | [
"Inverse of Vandermonde Matrix"
] | [
"Viète's Formulas",
"Definition:Matrix/Square Matrix/Order",
"Definition:Inverse Matrix"
] | [
"Recursion Property of Elementary Symmetric Function"
] |
proofwiki-16661 | Linear First Order ODE/dy = f(x) dx | Let $f: \R \to \R$ be an integrable real function.
The linear first order ODE:
:$(1): \quad \dfrac {\d y} {\d x} = \map f x$
has the general solution:
:$y = \ds \int \map f x \rd x + C$
where $\ds \int \map f x \rd x$ denotes the primitive of $f$. | {{begin-eqn}}
{{eqn | l = \dfrac {\d y} {\d x}
| r = \map f x
| c =
}}
{{eqn | ll= \leadsto
| l = \int \d y
| r = \int \map f x \rd x
| c = Solution to Separable Differential Equation
}}
{{eqn | ll= \leadsto
| l = y
| r = \int \map f x \rd x
| c = Primitive of Constant
}... | Let $f: \R \to \R$ be an [[Definition:Integrable Function|integrable]] [[Definition:Real Function|real function]].
The [[Definition:Linear First Order ODE|linear first order ODE]]:
:$(1): \quad \dfrac {\d y} {\d x} = \map f x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = \d... | {{begin-eqn}}
{{eqn | l = \dfrac {\d y} {\d x}
| r = \map f x
| c =
}}
{{eqn | ll= \leadsto
| l = \int \d y
| r = \int \map f x \rd x
| c = [[Solution to Separable Differential Equation]]
}}
{{eqn | ll= \leadsto
| l = y
| r = \int \map f x \rd x
| c = [[Primitive of Cons... | Linear First Order ODE/dy = f(x) dx | https://proofwiki.org/wiki/Linear_First_Order_ODE/dy_=_f(x)_dx | https://proofwiki.org/wiki/Linear_First_Order_ODE/dy_=_f(x)_dx | [
"Examples of Linear First Order ODEs"
] | [
"Definition:Integrable Function",
"Definition:Real Function",
"Definition:Linear First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Primitive (Calculus)"
] | [
"Solution to Separable Differential Equation",
"Primitive of Constant"
] |
proofwiki-16662 | Linear First Order ODE/dy = f(x) dx/Initial Condition | Consider the linear first order ODE:
:$(1): \quad \dfrac {\d y} {\d x} = \map f x$
subject to the initial condition:
:$y = y_0$ when $x = x_0$
$(1)$ has the particular solution:
:$y = y_0 + \ds \int_{x_0}^x \map f \xi \rd \xi$
where $\ds \int \map f x \rd x$ denotes the primitive of $f$. | It is seen that $(1)$ is an instance of the first order ordinary differential equation:
:$\dfrac {\d y} {\d x} = \map f {x, y}$
which is:
:subject to an initial condition: $\tuple {x_0, y_0}$
where:
:$\map f {x, y}$ is actually $\map f x$
From Solution to First Order Initial Value Problem, this problem is equivalent to... | Consider the [[Definition:Linear First Order ODE|linear first order ODE]]:
:$(1): \quad \dfrac {\d y} {\d x} = \map f x$
subject to the [[Definition:Initial Condition|initial condition]]:
:$y = y_0$ when $x = x_0$
$(1)$ has the [[Definition:Particular Solution to Differential Equation|particular solution]]:
:$y = y_... | It is seen that $(1)$ is an instance of the [[Definition:First Order Ordinary Differential Equation|first order ordinary differential equation]]:
:$\dfrac {\d y} {\d x} = \map f {x, y}$
which is:
:subject to an [[Definition:Initial Condition|initial condition]]: $\tuple {x_0, y_0}$
where:
:$\map f {x, y}$ is actually $... | Linear First Order ODE/dy = f(x) dx/Initial Condition | https://proofwiki.org/wiki/Linear_First_Order_ODE/dy_=_f(x)_dx/Initial_Condition | https://proofwiki.org/wiki/Linear_First_Order_ODE/dy_=_f(x)_dx/Initial_Condition | [
"Examples of Linear First Order ODEs"
] | [
"Definition:Linear First Order Ordinary Differential Equation",
"Definition:Initial Condition",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Primitive (Calculus)"
] | [
"Definition:First Order Ordinary Differential Equation",
"Definition:Initial Condition",
"Solution to First Order Initial Value Problem",
"Definition:Integral Equation"
] |
proofwiki-16663 | Solution by Integrating Factor/Examples/y' - 3y = sin x | The linear first order ODE:
:$\dfrac {\d y} {\d x} - 3 y = \sin x$
has the general solution:
:$y = \dfrac 1 {10} \paren {3 \sin x - \cos x} + C e^{3 x}$ | This is a linear first order ODE in the form:
:$\dfrac {\d y} {\d x} + \map P x y = \map Q x$
where:
:$\map P x y = -3 y$
:$\map Q x = \sin x$
Thus:
{{begin-eqn}}
{{eqn | l = \int \map P x \rd x
| r = \int -3 \rd x
| c =
}}
{{eqn | r = -3 x
| c =
}}
{{eqn | ll= \leadsto
| l = e^{\int P \rd x}
... | The [[Definition:Linear First Order ODE|linear first order ODE]]:
:$\dfrac {\d y} {\d x} - 3 y = \sin x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = \dfrac 1 {10} \paren {3 \sin x - \cos x} + C e^{3 x}$ | This 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:
:$\map P x y = -3 y$
:$\map Q x = \sin x$
Thus:
{{begin-eqn}}
{{eqn | l = \int \map P x \rd x
| r = \int -3 \rd x
| c =
}}
{{eqn | r = -3 x
| c =
}}
{{eqn | ll... | Solution by Integrating Factor/Examples/y' - 3y = sin x/Proof 1 | https://proofwiki.org/wiki/Solution_by_Integrating_Factor/Examples/y'_-_3y_=_sin_x | https://proofwiki.org/wiki/Solution_by_Integrating_Factor/Examples/y'_-_3y_=_sin_x/Proof_1 | [
"Examples of Solution by Integrating Factor",
"Solution by Integrating Factor/Examples/y' - 3y"
] | [
"Definition:Linear First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Linear First Order Ordinary Differential Equation",
"Solution to Linear First Order Ordinary Differential Equation/Solution by Integrating Factor",
"Primitive of Exponential of a x by Sine of b x"
] |
proofwiki-16664 | Solution by Integrating Factor/Examples/y' - 3y = sin x | The linear first order ODE:
:$\dfrac {\d y} {\d x} - 3 y = \sin x$
has the general solution:
:$y = \dfrac 1 {10} \paren {3 \sin x - \cos x} + C e^{3 x}$ | This is a linear first order ODE with constant coefficents in the form:
:$\dfrac {\d y} {\d x} + a y = \map Q x$
where:
:$a = -3$
:$\map Q x = \sin x$
Thus from Solution to Linear First Order ODE with Constant Coefficients:
{{begin-eqn}}
{{eqn | l = y
| r = e^{-3 x} \int e^{3 x} \sin x \rd x + C e^{-3 x}
| ... | The [[Definition:Linear First Order ODE|linear first order ODE]]:
:$\dfrac {\d y} {\d x} - 3 y = \sin x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = \dfrac 1 {10} \paren {3 \sin x - \cos x} + C e^{3 x}$ | This is a [[Definition:Linear First Order ODE with Constant Coefficients|linear first order ODE with constant coefficents]] in the form:
:$\dfrac {\d y} {\d x} + a y = \map Q x$
where:
:$a = -3$
:$\map Q x = \sin x$
Thus from [[Solution to Linear First Order ODE with Constant Coefficients]]:
{{begin-eqn}}
{{eqn | l =... | Solution by Integrating Factor/Examples/y' - 3y = sin x/Proof 2 | https://proofwiki.org/wiki/Solution_by_Integrating_Factor/Examples/y'_-_3y_=_sin_x | https://proofwiki.org/wiki/Solution_by_Integrating_Factor/Examples/y'_-_3y_=_sin_x/Proof_2 | [
"Examples of Solution by Integrating Factor",
"Solution by Integrating Factor/Examples/y' - 3y"
] | [
"Definition:Linear First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Linear First Order Ordinary Differential Equation/Constant Coefficients",
"Solution to Linear First Order ODE with Constant Coefficients",
"Primitive of Exponential of a x by Sine of b x"
] |
proofwiki-16665 | Solution to Linear First Order ODE with Constant Coefficients | A linear first order ODE with constant coefficients in the form:
:$(1): \quad \dfrac {\d y} {\d x} + a y = \map Q x$
has the general solution:
:$\ds y = e^{-a x} \paren {\int e^{a x} \map Q x \rd x + C}$ | From the Product Rule for Derivatives:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {e^{a x} y}
| r = e^{a x} \cdot \dfrac {\d y} {\d x} + y \cdot a e^{a x}
| c =
}}
{{eqn | r = e^{a x} \paren {\dfrac {\d y} {\d x} + a y}
| c =
}}
{{end-eqn}}
Hence, multiplying $(1)$ all through by $e^{\int a \... | A [[Definition:Linear First Order ODE with Constant Coefficients|linear first order ODE with constant coefficients]] in the form:
:$(1): \quad \dfrac {\d y} {\d x} + a y = \map Q x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$\ds y = e^{-a x} \paren {\int e^{a x} \map Q x \rd x... | From the [[Product Rule for Derivatives]]:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {e^{a x} y}
| r = e^{a x} \cdot \dfrac {\d y} {\d x} + y \cdot a e^{a x}
| c =
}}
{{eqn | r = e^{a x} \paren {\dfrac {\d y} {\d x} + a y}
| c =
}}
{{end-eqn}}
Hence, multiplying $(1)$ all through by $e^{\... | Solution to Linear First Order ODE with Constant Coefficients/Proof 1 | https://proofwiki.org/wiki/Solution_to_Linear_First_Order_ODE_with_Constant_Coefficients | https://proofwiki.org/wiki/Solution_to_Linear_First_Order_ODE_with_Constant_Coefficients/Proof_1 | [
"Linear First Order ODEs with Constant Coefficients",
"Solution to Linear First Order ODE with Constant Coefficients"
] | [
"Definition:Linear First Order Ordinary Differential Equation/Constant Coefficients",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Product Rule for Derivatives"
] |
proofwiki-16666 | Solution to Linear First Order ODE with Constant Coefficients | A linear first order ODE with constant coefficients in the form:
:$(1): \quad \dfrac {\d y} {\d x} + a y = \map Q x$
has the general solution:
:$\ds y = e^{-a x} \paren {\int e^{a x} \map Q x \rd x + C}$ | This is a specific instance of Solution to Linear First Order Ordinary Differential Equation:
{{:Solution to Linear First Order Ordinary Differential Equation}}
In this instance, we have:
:$\map P x = a$
Hence:
:$\ds \int \map P x = a x$
and the result follows.
{{qed}} | A [[Definition:Linear First Order ODE with Constant Coefficients|linear first order ODE with constant coefficients]] in the form:
:$(1): \quad \dfrac {\d y} {\d x} + a y = \map Q x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$\ds y = e^{-a x} \paren {\int e^{a x} \map Q x \rd x... | This is a specific instance of [[Solution to Linear First Order Ordinary Differential Equation]]:
{{:Solution to Linear First Order Ordinary Differential Equation}}
In this instance, we have:
:$\map P x = a$
Hence:
:$\ds \int \map P x = a x$
and the result follows.
{{qed}} | Solution to Linear First Order ODE with Constant Coefficients/Proof 2 | https://proofwiki.org/wiki/Solution_to_Linear_First_Order_ODE_with_Constant_Coefficients | https://proofwiki.org/wiki/Solution_to_Linear_First_Order_ODE_with_Constant_Coefficients/Proof_2 | [
"Linear First Order ODEs with Constant Coefficients",
"Solution to Linear First Order ODE with Constant Coefficients"
] | [
"Definition:Linear First Order Ordinary Differential Equation/Constant Coefficients",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Solution to Linear First Order Ordinary Differential Equation"
] |
proofwiki-16667 | Solution to Linear First Order ODE with Constant Coefficients/With Initial Condition | Consider the linear first order ODE with constant coefficients in the form:
:$(1): \quad \dfrac {\d y} {\d x} + a y = \map Q x$
with initial condition $\tuple {x_0, y_0}$
Then $(1)$ has the particular solution:
:$\ds y = e^{-a x} \int_{x_0}^x e^{a \xi} \map Q \xi \rd \xi + y_0 e^{a \paren {x - x_0} }$ | From Solution to Linear First Order ODE with Constant Coefficients, the general solution to $(1)$ is:
:$(2): \quad \ds y = e^{-a x} \int e^{a x} \map Q x \rd x + C e^{-a x}$
Let $y = y_0$ when $x = x_0$.
We have:
:$(3): \ds \quad y_0 = e^{-a x_0} \int e^{a x_0} \map Q {x_0} \rd x_0 + C e^{-a x_0}$
Thus:
{{begin-eqn}}
{... | Consider the [[Definition:Linear First Order ODE with Constant Coefficients|linear first order ODE with constant coefficients]] in the form:
:$(1): \quad \dfrac {\d y} {\d x} + a y = \map Q x$
with [[Definition:Initial Condition|initial condition]] $\tuple {x_0, y_0}$
Then $(1)$ has the [[Definition:Particular Solutio... | From [[Solution to Linear First Order ODE with Constant Coefficients]], the [[Definition:General Solution to Differential Equation|general solution]] to $(1)$ is:
:$(2): \quad \ds y = e^{-a x} \int e^{a x} \map Q x \rd x + C e^{-a x}$
Let $y = y_0$ when $x = x_0$.
We have:
:$(3): \ds \quad y_0 = e^{-a x_0} \int e^{... | Solution to Linear First Order ODE with Constant Coefficients/With Initial Condition | https://proofwiki.org/wiki/Solution_to_Linear_First_Order_ODE_with_Constant_Coefficients/With_Initial_Condition | https://proofwiki.org/wiki/Solution_to_Linear_First_Order_ODE_with_Constant_Coefficients/With_Initial_Condition | [
"Linear First Order ODEs with Constant Coefficients"
] | [
"Definition:Linear First Order Ordinary Differential Equation/Constant Coefficients",
"Definition:Initial Condition",
"Definition:Differential Equation/Solution/Particular Solution"
] | [
"Solution to Linear First Order ODE with Constant Coefficients",
"Definition:Differential Equation/Solution/General Solution",
"Fundamental Theorem of Calculus"
] |
proofwiki-16668 | Solution by Integrating Factor/Examples/y' + y = x^-1 | Consider the linear first order ODE:
:$(1): \quad \dfrac {\d y} {\d x} + y = \dfrac 1 x$
with the initial condition $\tuple {1, 0}$.
This has the particular solution:
:$y = \ds e^{-x} \int_1^x \dfrac {e^\xi \rd \xi} \xi$ | This is a linear first order ODE with constant coefficents in the form:
:$\dfrac {\d y} {\d x} + a y = \map Q x$
where:
:$a = 1$
:$\map Q x = \dfrac 1 x$
with the initial condition $y = 0$ when $x = 1$.
Thus from Solution to Linear First Order ODE with Constant Coefficients with Initial Condition:
{{begin-eqn}}
{{eqn |... | Consider the [[Definition:Linear First Order ODE|linear first order ODE]]:
:$(1): \quad \dfrac {\d y} {\d x} + y = \dfrac 1 x$
with the [[Definition:Initial Condition|initial condition]] $\tuple {1, 0}$.
This has the [[Definition:Particular Solution of Differential Equation|particular solution]]:
:$y = \ds e^{-x} \int... | This is a [[Definition:Linear First Order ODE with Constant Coefficients|linear first order ODE with constant coefficents]] in the form:
:$\dfrac {\d y} {\d x} + a y = \map Q x$
where:
:$a = 1$
:$\map Q x = \dfrac 1 x$
with the [[Definition:Initial Condition|initial condition]] $y = 0$ when $x = 1$.
Thus from [[Solut... | Solution by Integrating Factor/Examples/y' + y = x^-1 | https://proofwiki.org/wiki/Solution_by_Integrating_Factor/Examples/y'_+_y_=_x^-1 | https://proofwiki.org/wiki/Solution_by_Integrating_Factor/Examples/y'_+_y_=_x^-1 | [
"Examples of Solution by Integrating Factor"
] | [
"Definition:Linear First Order Ordinary Differential Equation",
"Definition:Initial Condition",
"Definition:Differential Equation/Solution/Particular Solution"
] | [
"Definition:Linear First Order Ordinary Differential Equation/Constant Coefficients",
"Definition:Initial Condition",
"Solution to Linear First Order ODE with Constant Coefficients/With Initial Condition",
"Primitive of exp x over x has no Solution in Elementary Functions"
] |
proofwiki-16669 | General Solution equals Particular Solution plus Complementary Function | Consider the linear first order ODE with constant coefficients:
:$(1): \quad \dfrac {\d y} {\d x} + a y = \map Q x$
The general solution to $(1)$ consists of:
:the particular solution to $(1)$ for which the constant of integration is $0$
plus:
:the complementary function to $(1)$. | From Solution to Linear First Order ODE with Constant Coefficients, $(1)$ has the general solution:
:$\ds y = e^{-a x} \int e^{a x} \map Q x \rd x + C e^{-a x}$
Setting $C = 0$ we get:
:$\ds y = e^{-a x} \int e^{a x} \map Q x \rd x$
which is a particular solution to $(1)$
By definition, the complementary function to $(... | Consider the [[Definition:Linear First Order ODE with Constant Coefficients|linear first order ODE with constant coefficients]]:
:$(1): \quad \dfrac {\d y} {\d x} + a y = \map Q x$
The [[Definition:General Solution to Differential Equation|general solution]] to $(1)$ consists of:
:the [[Definition:Particular Solution... | From [[Solution to Linear First Order ODE with Constant Coefficients]], $(1)$ has the [[Definition:General Solution to Differential Equation|general solution]]:
:$\ds y = e^{-a x} \int e^{a x} \map Q x \rd x + C e^{-a x}$
Setting $C = 0$ we get:
:$\ds y = e^{-a x} \int e^{a x} \map Q x \rd x$
which is a [[Definiti... | General Solution equals Particular Solution plus Complementary Function | https://proofwiki.org/wiki/General_Solution_equals_Particular_Solution_plus_Complementary_Function | https://proofwiki.org/wiki/General_Solution_equals_Particular_Solution_plus_Complementary_Function | [
"Solution to Linear First Order ODE with Constant Coefficients"
] | [
"Definition:Linear First Order Ordinary Differential Equation/Constant Coefficients",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Primitive (Calculus)/Constant of Integration",
"Definition:Complementary Function o... | [
"Solution to Linear First Order ODE with Constant Coefficients",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Complementary Function of Linear First Order ODE With Constant Coefficients",
"Definition:Differential E... |
proofwiki-16670 | Derivation of Auxiliary Equation to Constant Coefficient LSOODE | Consider the linear second order ODE with constant coefficients:
:$(1): \quad \dfrac {\d^2 y} {\d x^2} + p \dfrac {\d y} {\d x} + q y = \map R x$
and its auxiliary equation:
:$(2): \quad m^2 + p m + q = 0$
The fact that the solutions of $(2)$ dictate the general solution of $(1)$ can be derived. | Let the reduced equation of $(1)$ be expressed in the form:
:$(3): \quad D^2 y + p D y + q y = 0$
where $D$ denotes the derivative operator {{WRT|Differentiation}} $x$:
:$D := \dfrac \d {\d x}$
Thus:
:$D^2 := \dfrac {\d^2} {\d x^2}$
We can express $(3)$ in the form:
:$(4): \quad \paren {D^2 + p y + q} y = 0$
Consider t... | Consider the [[Definition:Linear Second Order ODE with Constant Coefficients|linear second order ODE with constant coefficients]]:
:$(1): \quad \dfrac {\d^2 y} {\d x^2} + p \dfrac {\d y} {\d x} + q y = \map R x$
and its [[Definition:Auxiliary Equation|auxiliary equation]]:
:$(2): \quad m^2 + p m + q = 0$
The fact ... | Let the [[Definition:Reduced Equation of Linear Second Order ODE With Constant Coefficients|reduced equation]] of $(1)$ be expressed in the form:
:$(3): \quad D^2 y + p D y + q y = 0$
where $D$ denotes the [[Definition:Derivative|derivative operator]] {{WRT|Differentiation}} $x$:
:$D := \dfrac \d {\d x}$
Thus:
:$D^2 ... | Derivation of Auxiliary Equation to Constant Coefficient LSOODE | https://proofwiki.org/wiki/Derivation_of_Auxiliary_Equation_to_Constant_Coefficient_LSOODE | https://proofwiki.org/wiki/Derivation_of_Auxiliary_Equation_to_Constant_Coefficient_LSOODE | [
"Constant Coefficient LSOODEs"
] | [
"Definition:Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Reduced Equation of Linear ODE with Constant Coefficients/Second Order",
"Definition:Derivative",
"Definition:Constant",
"Definition:Real Number",
"Sum of Roots of Quadratic Equation",
"Product of Roots of Quadratic Equation",
"Solution to Linear First Order ODE with Constant Coefficients",
... |
proofwiki-16671 | Linear Second Order ODE/y'' + 2 y' + 2 y = 0 | The second order ODE:
:$(1): \quad y' ' + 2 y' + 2 y = 0$
has the general solution:
:$y = e^{-x} \paren {A \cos x + B \sin x}$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad: m^2 + 2 m + 2 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = -1 + i$
:$m_2 = -1 - i$
These are complex conjugates.
So from Solution of Const... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' + 2 y' + 2 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = e^{-x} \paren {A \cos x + B \sin x}$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad: m^2 + 2 m + 2 = 0$
From [[Solution to Quadratic Equation with Real Coefficients]],... | Linear Second Order ODE/y'' + 2 y' + 2 y = 0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_2_y'_+_2_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_2_y'_+_2_y_=_0 | [
"Examples of Constant Coefficient Homogeneous LSOODEs",
"Linear Second Order ODE/y'' + 2 y' + 2 y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Definition:Complex Conjugate",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differenti... |
proofwiki-16672 | Linear Second Order ODE/y'' + 2 y' + 2 y = 0/Verification | The equation:
:$(1): \quad y = e^{-x} \paren {A \cos x + B \sin x}$
is a set of solutions to the second order ODE:
:$y' ' + 2 y' + 2 y = 0$ | Differentiating $(1)$ twice {{WRT|Differentiation}} $x$ gives:
{{begin-eqn}}
{{eqn | l = y'
| r = -e^{-x} \paren {A \cos x + B \sin x} + e^{-x} \paren {-A \sin x + B \cos x}
| c =
}}
{{eqn | r = e^{-x} \paren {\paren {B - A} \cos x - \paren {A + B} \sin x}
| c = rearranging
}}
{{eqn | ll= \leadsto
... | The equation:
:$(1): \quad y = e^{-x} \paren {A \cos x + B \sin x}$
is a [[Definition:Set|set]] of [[Definition:Solution to Differential Equation|solutions]] to the [[Definition:Linear Second Order ODE|second order ODE]]:
:$y' ' + 2 y' + 2 y = 0$ | [[Definition:Differentiation|Differentiating]] $(1)$ twice {{WRT|Differentiation}} $x$ gives:
{{begin-eqn}}
{{eqn | l = y'
| r = -e^{-x} \paren {A \cos x + B \sin x} + e^{-x} \paren {-A \sin x + B \cos x}
| c =
}}
{{eqn | r = e^{-x} \paren {\paren {B - A} \cos x - \paren {A + B} \sin x}
| c = rearra... | Linear Second Order ODE/y'' + 2 y' + 2 y = 0/Verification | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_2_y'_+_2_y_=_0/Verification | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_2_y'_+_2_y_=_0/Verification | [
"Linear Second Order ODE/y'' + 2 y' + 2 y = 0"
] | [
"Definition:Set",
"Definition:Differential Equation/Solution",
"Definition:Linear Second Order Ordinary Differential Equation"
] | [
"Definition:Differentiation",
"Definition:Linear Second Order Ordinary Differential Equation"
] |
proofwiki-16673 | Linear Second Order ODE/y'' - 7 y' - 5 y = x^3 - 1 | The second order ODE:
:$(1): \quad y'' - 7 y' - 5 y = x^3 - 1$
has the general solution:
:$y = C_1 \, \map \exp {\paren {\dfrac 7 2 + \dfrac {\sqrt {69} } 2} x} + C_2 \, \map \exp {\paren {\dfrac 7 2 - \dfrac {\sqrt {69} } 2} x} + \dfrac 1 {625} \paren {-125 x^3 + 525 x^2 - 1620 x + 2603}$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = -7$
:$q = -5$
:$\map R x = x^3 - 1$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$(2): \quad y'' - ... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' - 7 y' - 5 y = x^3 - 1$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 \, \map \exp {\paren {\dfrac 7 2 + \dfrac {\sqrt {69} } 2} x} + C_2 \, \map \exp {\paren {\dfrac 7 2 - \dfrac {\sqrt {69} } 2} x} +... | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] with [[Definition:Constant|constant]] [[Definition:Coefficient|coefficients]] in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = -7$
:$q = -5$
:$\map R x = x^3 - 1$
First we establish the ... | Linear Second Order ODE/y'' - 7 y' - 5 y = x^3 - 1 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_7_y'_-_5_y_=_x^3_-_1 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_7_y'_-_5_y_=_x^3_-_1 | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Undetermined Coefficients"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Constant",
"Definition:Coefficient",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' - 7 y' - 5 y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Dif... |
proofwiki-16674 | Linear Second Order ODE/y'' - 7 y' - 5 y = 0 | The second order ODE:
:$(1): \quad y' ' - 7 y' - 5 y = 0$
has the general solution:
:$y = C_1 \, \map \exp {\paren {\dfrac 7 2 + \dfrac {\sqrt {69} } 2} x} + C_2 \, \map \exp {\paren {\dfrac 7 2 - \dfrac {\sqrt {69} } 2} x}$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad: m^2 - 7 m - 5 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
{{begin-eqn}}
{{eqn | l = m
| r = \dfrac {7 \pm \sqrt {7^2 - 4 \times 1 \times \p... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' - 7 y' - 5 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 \, \map \exp {\paren {\dfrac 7 2 + \dfrac {\sqrt {69} } 2} x} + C_2 \, \map \exp {\paren {\dfrac 7 2 - \dfrac {\sqrt {69} } 2} x}$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad: m^2 - 7 m - 5 = 0$
From [[Solution to Quadratic Equation with Real Coefficients]],... | Linear Second Order ODE/y'' - 7 y' - 5 y = 0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_7_y'_-_5_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_7_y'_-_5_y_=_0 | [
"Examples of Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Definition:Real Number",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential Equ... |
proofwiki-16675 | Convergent Sequence is Cauchy Sequence/Normed Vector Space | Let $\struct{X, \norm{\,\cdot\,} }$ be a normed vector space.
Every convergent sequence in $X$ is a Cauchy sequence. | Let $\sequence {x_n}$ be a sequence in $X$ that converges to the limit $L \in X$.
Let $\epsilon > 0$.
Then also $\dfrac \epsilon 2 > 0$.
Because $\sequence {x_n}$ converges to $L$, we have:
:$\exists N: \forall n > N: \norm {x_n - L} < \dfrac \epsilon 2$
So if $m > N$ and $n > N$, then:
{{begin-eqn}}
{{eqn | l = \norm... | Let $\struct{X, \norm{\,\cdot\,} }$ be a [[Definition:Normed Vector Space|normed vector space]].
Every [[Definition:Convergent Sequence in Normed Vector Space|convergent sequence]] in $X$ is a [[Definition:Cauchy Sequence in Normed Vector Space|Cauchy sequence]]. | Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence]] in $X$ that [[Definition:Convergent Sequence in Normed Vector Space|converges]] to the [[Definition:Limit of Sequence in Normed Vector Space|limit]] $L \in X$.
Let $\epsilon > 0$.
Then also $\dfrac \epsilon 2 > 0$.
Because $\sequence {x_n}$ [[Definition:Co... | Convergent Sequence is Cauchy Sequence/Normed Vector Space | https://proofwiki.org/wiki/Convergent_Sequence_is_Cauchy_Sequence/Normed_Vector_Space | https://proofwiki.org/wiki/Convergent_Sequence_is_Cauchy_Sequence/Normed_Vector_Space | [
"Cauchy Sequences",
"Convergent Sequence is Cauchy Sequence",
"Convergent Sequences (Normed Vector Spaces)",
"Convergent Sequence is Cauchy Sequence"
] | [
"Definition:Normed Vector Space",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Cauchy Sequence/Normed Vector Space"
] | [
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Limit of Sequence/Normed Vector Space",
"Definition:Convergent Sequence/Metric Space",
"Definition:Norm/Vector Space",
"Definition:Cauchy Sequence/Normed Vector Space"
] |
proofwiki-16676 | Linear Second Order ODE/y'' - 3 y' + 2 y = 5 exp 3 x | The second order ODE:
:$(1): \quad y'' - 3 y' + 2 y = 5 e^{3 x}$
has the general solution:
:$y = C_1 e^x + C_2 e^{2 x} + \dfrac {5 e^{3 x} } 2$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = -3$
:$q = 2$
:$\map R x = 5 e^{3 x}$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$y'' - 3 y' + 2 y... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' - 3 y' + 2 y = 5 e^{3 x}$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^x + C_2 e^{2 x} + \dfrac {5 e^{3 x} } 2$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] with [[Definition:Constant|constant]] [[Definition:Coefficient|coefficients]] in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = -3$
:$q = 2$
:$\map R x = 5 e^{3 x}$
First we establish the... | Linear Second Order ODE/y'' - 3 y' + 2 y = 5 exp 3 x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_3_y'_+_2_y_=_5_exp_3_x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_3_y'_+_2_y_=_5_exp_3_x | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Undetermined Coefficients"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Constant",
"Definition:Coefficient",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' - 3 y' + 2 y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Method of Unde... |
proofwiki-16677 | Linear Second Order ODE/y'' - y = 3 exp -x | The second order ODE:
:$(1): \quad y'' - y = 3 e^{-x}$
has the general solution:
:$y = C_1 e^x + C_2 e^{-x} - \dfrac {3 x e^{-x} } 2$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 0$
:$q = -1$
:$\map R x = 3 e^{-x}$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$y'' - y = 0$
From... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' - y = 3 e^{-x}$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^x + C_2 e^{-x} - \dfrac {3 x e^{-x} } 2$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] with [[Definition:Constant|constant]] [[Definition:Coefficient|coefficients]] in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 0$
:$q = -1$
:$\map R x = 3 e^{-x}$
First we establish the ... | Linear Second Order ODE/y'' - y = 3 exp -x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_y_=_3_exp_-x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_y_=_3_exp_-x | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Undetermined Coefficients"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Constant",
"Definition:Coefficient",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' - y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Method of Undetermined ... |
proofwiki-16678 | Linear Second Order ODE/y'' - 2 y' + y = exp x | The second order ODE:
:$(1): \quad y'' - 2 y' + y = e^x$
has the general solution:
:$y = C_1 e^x + C_2 x e^x + \dfrac {x^2 e^x} 2$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = -2$
:$q = 1$
:$\map R x = e^x$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$y'' - 2 y' + y = 0$
Fr... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' - 2 y' + y = e^x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^x + C_2 x e^x + \dfrac {x^2 e^x} 2$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] with [[Definition:Constant|constant]] [[Definition:Coefficient|coefficients]] in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = -2$
:$q = 1$
:$\map R x = e^x$
First we establish the solut... | Linear Second Order ODE/y'' - 2 y' + y = exp x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_+_y_=_exp_x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_+_y_=_exp_x | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Undetermined Coefficients"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Constant",
"Definition:Coefficient",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' - 2 y' + y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Method of Undete... |
proofwiki-16679 | Linear Second Order ODE/y'' - 2 y' - 5 y = 0 | The second order ODE:
:$(1): \quad y' ' - 2 y' - 5 y = 0$
has the general solution:
:$y = C_1 \map \exp {\paren {1 + \sqrt 6} x} + C_2 \map \exp {\paren {1 - \sqrt 6} x}$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad: m^2 - 2 m - 5 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = 1 + \sqrt 6$
:$m_2 = 1 - \sqrt 6$
So from Solution of Constant Coefficient Homo... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' - 2 y' - 5 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 \map \exp {\paren {1 + \sqrt 6} x} + C_2 \map \exp {\paren {1 - \sqrt 6} x}$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad: m^2 - 2 m - 5 = 0$
From [[Solution to Quadratic Equation with Real Coefficients]]... | Linear Second Order ODE/y'' - 2 y' - 5 y = 0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_-_5_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_-_5_y_=_0 | [
"Examples of Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential Equation/Solution/General Solut... |
proofwiki-16680 | Linear Second Order ODE/y'' - 2 y' - 5 y = 2 cos 3 x - sin 3 x | The second order ODE:
:$(1): \quad y'' - 2 y' - 5 y = 2 \cos 3 x - \sin 3 x$
has the general solution:
:$y = C_1 \map \exp {\paren {1 + \sqrt 6} x} + C_2 \map \exp {\paren {1 - \sqrt 6} x} + \dfrac 1 {116} \paren {\sin 3 x - 17 \cos 3 x}$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = -2$
:$q = -5$
:$\map R x = 2 \cos 3 x - \sin 3 x$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$y''... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' - 2 y' - 5 y = 2 \cos 3 x - \sin 3 x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 \map \exp {\paren {1 + \sqrt 6} x} + C_2 \map \exp {\paren {1 - \sqrt 6} x} + \dfrac 1 {116} \paren {\sin 3 x - 17 \c... | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] with [[Definition:Constant|constant]] [[Definition:Coefficient|coefficients]] in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = -2$
:$q = -5$
:$\map R x = 2 \cos 3 x - \sin 3 x$
First we ... | Linear Second Order ODE/y'' - 2 y' - 5 y = 2 cos 3 x - sin 3 x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_-_5_y_=_2_cos_3_x_-_sin_3_x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_-_5_y_=_2_cos_3_x_-_sin_3_x | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Undetermined Coefficients",
"Linear Second Order ODE/y'' - 2 y' - 5 y = 2 cos 3 x - sin 3 x"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Constant",
"Definition:Coefficient",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' - 2 y' - 5 y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Dif... |
proofwiki-16681 | Linear Second Order ODE/y'' - 2 y' - 5 y = 2 cos 3 x - sin 3 x/Particular Solution | The second order ODE:
:$(1): \quad y'' - 2 y' - 5 y = 2 \cos 3 x - \sin 3 x$
has a particular solution:
:$y_p = \dfrac 1 {116} \paren {\sin 3 x - 17 \cos 3 x}$ | From Linear Second Order ODE: $y'' - 2 y' - 5 y = 0$, we have established that the general solution to $(1)$ is:
:$y_g = C_1 \map \exp {\paren {1 + \sqrt 6} x} + C_2 \map \exp {\paren {1 - \sqrt 6} x}$ | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' - 2 y' - 5 y = 2 \cos 3 x - \sin 3 x$
has a [[Definition:Particular Solution to Differential Equation|particular solution]]:
:$y_p = \dfrac 1 {116} \paren {\sin 3 x - 17 \cos 3 x}$ | From [[Linear Second Order ODE/y'' - 2 y' - 5 y = 0|Linear Second Order ODE: $y'' - 2 y' - 5 y = 0$]], we have established that the [[Definition:General Solution to Differential Equation|general solution]] to $(1)$ is:
:$y_g = C_1 \map \exp {\paren {1 + \sqrt 6} x} + C_2 \map \exp {\paren {1 - \sqrt 6} x}$ | Linear Second Order ODE/y'' - 2 y' - 5 y = 2 cos 3 x - sin 3 x/Particular Solution | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_-_5_y_=_2_cos_3_x_-_sin_3_x/Particular_Solution | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_-_5_y_=_2_cos_3_x_-_sin_3_x/Particular_Solution | [
"Linear Second Order ODE/y'' - 2 y' - 5 y = 2 cos 3 x - sin 3 x",
"Examples of Method of Undetermined Coefficients"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/Particular Solution"
] | [
"Linear Second Order ODE/y'' - 2 y' - 5 y = 0",
"Definition:Differential Equation/Solution/General Solution"
] |
proofwiki-16682 | Linear Second Order ODE/y'' + 4 y = 3 sin 2 x | The second order ODE:
:$(1): \quad y'' + 4 y = 3 \sin 2 x$
has the general solution:
:$y = C_1 \sin k x + C_2 \cos k x - \dfrac 3 4 x \cos 2 x$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 0$
:$q = 4$
:$\map R x = 3 \sin 2 x$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$(2): \quad y'' +... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' + 4 y = 3 \sin 2 x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 \sin k x + C_2 \cos k x - \dfrac 3 4 x \cos 2 x$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] with [[Definition:Constant|constant]] [[Definition:Coefficient|coefficients]] in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 0$
:$q = 4$
:$\map R x = 3 \sin 2 x$
First we establish the... | Linear Second Order ODE/y'' + 4 y = 3 sin 2 x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_4_y_=_3_sin_2_x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_4_y_=_3_sin_2_x | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Undetermined Coefficients"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Constant",
"Definition:Coefficient",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' + 4 y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Differenti... |
proofwiki-16683 | Cauchy Sequence is Bounded/Normed Vector Space | Let $V = \struct {X, \norm {\,\cdot\,} }$ be a normed vector space.
Every Cauchy sequence in $X$ is bounded. | Let $\sequence {x_n} $ be a Cauchy sequence in $V$.
Then by definition:
:$\forall \epsilon \in \R_{>0}: \exists N \in \N : \forall n, m \ge N: \norm {x_n - x_m} < \epsilon$
Let $N$ satisfy:
:$\forall n, m \ge N: \norm {x_n - x_m} < 1$
Let $m = N + 1 > N$.
Then $\forall n \ge N$:
{{begin-eqn}}
{{eqn | l = \norm {x_n}
... | Let $V = \struct {X, \norm {\,\cdot\,} }$ be a [[Definition:Normed Vector Space|normed vector space]].
Every [[Definition:Cauchy Sequence in Normed Vector Space|Cauchy sequence in $X$]] is [[Definition:Bounded Sequence in Normed Vector Space|bounded]]. | Let $\sequence {x_n} $ be a [[Definition:Cauchy Sequence in Normed Vector Space|Cauchy sequence in $V$]].
Then by definition:
:$\forall \epsilon \in \R_{>0}: \exists N \in \N : \forall n, m \ge N: \norm {x_n - x_m} < \epsilon$
Let $N$ satisfy:
:$\forall n, m \ge N: \norm {x_n - x_m} < 1$
Let $m = N + 1 > N$.
Then $... | Cauchy Sequence is Bounded/Normed Vector Space | https://proofwiki.org/wiki/Cauchy_Sequence_is_Bounded/Normed_Vector_Space | https://proofwiki.org/wiki/Cauchy_Sequence_is_Bounded/Normed_Vector_Space | [
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Cauchy Sequence/Normed Vector Space",
"Definition:Bounded Sequence/Normed Vector Space"
] | [
"Definition:Cauchy Sequence/Normed Vector Space",
"Definition:Bounded Sequence/Normed Vector Space"
] |
proofwiki-16684 | Linear Second Order ODE/y'' - 4 y' - 5 y = 0 | The second order ODE:
:$(1): \quad y' ' - 4 y' - 5 y = 0$
has the general solution:
:$y = C_1 e^{5 x} + C_2 e^{-x}$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad: m^2 - 4 m - 5 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = 5$
:$m_2 = -1$
These are real and unequal.
So from Solution of Constant Coeffic... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' - 4 y' - 5 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^{5 x} + C_2 e^{-x}$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad: m^2 - 4 m - 5 = 0$
From [[Solution to Quadratic Equation with Real Coefficients]]... | Linear Second Order ODE/y'' - 4 y' - 5 y = 0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_4_y'_-_5_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_4_y'_-_5_y_=_0 | [
"Examples of Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Definition:Real Number",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential Equ... |
proofwiki-16685 | Linear Second Order ODE/y'' - 4 y' - 5 y = x^2/y(0) = 1, y'(0) = -1 | Consider the second order ODE:
:$(1): \quad y'' - 4 y' - 5 y = x^2$
whose initial conditions are:
:$y = 1$ when $x = 0$
:$y' = -1$ when $x = 0$
$(1)$ has the particular solution:
:$y = \dfrac {e^{5 x} } {375} + \dfrac {4 e^{-x} } 3 - \dfrac {x^2} 5 + \dfrac {8 x} {25} - \dfrac {42} {125}$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = -4$
:$q = -5$
:$\map R x = x^2$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$(2): \quad y'' - 4 y'... | Consider the [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' - 4 y' - 5 y = x^2$
whose [[Definition:Initial Condition|initial conditions]] are:
:$y = 1$ when $x = 0$
:$y' = -1$ when $x = 0$
$(1)$ has the [[Definition:Particular Solution to Differential Equation|particular solution]]:
:$y = \dfrac ... | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] with [[Definition:Constant|constant]] [[Definition:Coefficient|coefficients]] in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = -4$
:$q = -5$
:$\map R x = x^2$
First we establish the solu... | Linear Second Order ODE/y'' - 4 y' - 5 y = x^2/y(0) = 1, y'(0) = -1 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_4_y'_-_5_y_=_x^2/y(0)_=_1,_y'(0)_=_-1 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_4_y'_-_5_y_=_x^2/y(0)_=_1,_y'(0)_=_-1 | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Undetermined Coefficients"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Initial Condition",
"Definition:Differential Equation/Solution/Particular Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Constant",
"Definition:Coefficient",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' - 4 y' - 5 y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Dif... |
proofwiki-16686 | Linear Second Order ODE/y'' - 2 y' + y = 1 over 1 + e^x | The second order ODE:
:$(1): \quad y'' - 2 y' + y = \dfrac 1 {1 + e^x}$
has a particular solution:
:$y = 1 + e^x \ds \int \map \ln {1 + e^{-x} } \rd x$ | {{begin-eqn}}
{{eqn | l = \paren {D^2 - 2 D + 1} y
| r = \dfrac 1 {1 + e^x}
| c = expressing $(1)$ in a different form
}}
{{eqn | ll= \leadsto
| l = \paren {D^2 - 1}^2 y
| r = \dfrac 1 {1 + e^x}
| c = and in a different form again
}}
{{eqn | ll= \leadsto
| l = \paren {D - 1} z
... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' - 2 y' + y = \dfrac 1 {1 + e^x}$
has a [[Definition:Particular Solution to Differential Equation|particular solution]]:
:$y = 1 + e^x \ds \int \map \ln {1 + e^{-x} } \rd x$ | {{begin-eqn}}
{{eqn | l = \paren {D^2 - 2 D + 1} y
| r = \dfrac 1 {1 + e^x}
| c = expressing $(1)$ in a different form
}}
{{eqn | ll= \leadsto
| l = \paren {D^2 - 1}^2 y
| r = \dfrac 1 {1 + e^x}
| c = and in a different form again
}}
{{eqn | ll= \leadsto
| l = \paren {D - 1} z
... | Linear Second Order ODE/y'' - 2 y' + y = 1 over 1 + e^x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_+_y_=_1_over_1_+_e^x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_+_y_=_1_over_1_+_e^x | [
"Examples of Constant Coefficient LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/Particular Solution"
] | [
"Solution to Linear First Order ODE with Constant Coefficients",
"Definition:Integrating Factor",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Primitive (Calculus)/Integration",
"Integration by Substitution",
"Primitive of Constant",
"Primitive of Reciprocal"
] |
proofwiki-16687 | ODE/(D^4 - 1) y = sin x | The second order ODE:
:$(1): \quad \paren {D^4 - 1} y' = \sin x$
has a general solution:
:$y = C_1 e^x + C_2 e^{-x} + C_3 \sin x + C_4 \cos x + \dfrac {x \cos x} 4$ | First we solve the reduced equation of $(1)$:
:$(2): \quad \paren {D^4 - 1} y' = 0$
The auxiliary equation of $(1)$ is:
:$(3): \quad: m^4 - 1 = 0$
From Complex $4$th Roots of Unity, the roots of $(2)$ are:
:$m_1 = 1$
:$m_2 = i$
:$m_3 = -1$
:$m_4 = -i$
So from Solution of Constant Coefficient Linear nth Order ODE, the g... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad \paren {D^4 - 1} y' = \sin x$
has a [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^x + C_2 e^{-x} + C_3 \sin x + C_4 \cos x + \dfrac {x \cos x} 4$ | First we solve the [[Definition:Reduced Equation of Linear ODE with Constant Coefficients|reduced equation]] of $(1)$:
:$(2): \quad \paren {D^4 - 1} y' = 0$
The [[Definition:Auxiliary Equation|auxiliary equation]] of $(1)$ is:
:$(3): \quad: m^4 - 1 = 0$
From [[Complex 4th Roots of Unity|Complex $4$th Roots of Unity]]... | ODE/(D^4 - 1) y = sin x | https://proofwiki.org/wiki/ODE/(D^4_-_1)_y_=_sin_x | https://proofwiki.org/wiki/ODE/(D^4_-_1)_y_=_sin_x | [
"Examples of Linear ODE"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Reduced Equation of Linear ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Complex Roots of Unity/Examples/4th Roots",
"Definition:Root of Polynomial",
"Solution of Constant Coefficient Linear nth Order ODE",
"Definition:Differential Equation/Solution/General Solution",
"D... |
proofwiki-16688 | P-adic Unit has Norm Equal to One | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\Z_p$ denote the $p$-adic integers.
Let $x \in \Q_p$.
Then x is a $p$-adic unit {{iff}} $\norm x_p = 1$ | === Necessary Condition ===
Let $x$ be a $p$-adic unit.
Then:
:$x \in \Z_p$
:$x^{-1} \in \Z_p$
By definition of the $p$-adic integers:
:$\norm x_p \le 1$
:$\norm {x^{-1} }_p \le 1$
From Norm of Inverse in Division Ring:
:$\norm x_p \ge 1$
It follows that:
:$\norm x_p = 1$
{{qed|lemma}} | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\Z_p$ denote the [[Definition:P-adic Integer|$p$-adic integers]].
Let $x \in \Q_p$.
Then x is a [[Definition:P-adic Unit|$p$-adic unit]] {{iff... | === Necessary Condition ===
Let $x$ be a [[Definition:P-adic Unit|$p$-adic unit]].
Then:
:$x \in \Z_p$
:$x^{-1} \in \Z_p$
By definition of the [[Definition:P-adic Integer|$p$-adic integers]]:
:$\norm x_p \le 1$
:$\norm {x^{-1} }_p \le 1$
From [[Norm of Inverse in Division Ring]]:
:$\norm x_p \ge 1$
It follows that... | P-adic Unit has Norm Equal to One | https://proofwiki.org/wiki/P-adic_Unit_has_Norm_Equal_to_One | https://proofwiki.org/wiki/P-adic_Unit_has_Norm_Equal_to_One | [
"P-adic Units"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:P-adic Integer",
"Definition:P-adic Unit"
] | [
"Definition:P-adic Unit",
"Definition:P-adic Integer",
"Properties of Norm on Division Ring/Norm of Inverse",
"Properties of Norm on Division Ring/Norm of Inverse",
"Definition:P-adic Integer",
"Definition:P-adic Unit"
] |
proofwiki-16689 | P-adic Number times P-adic Norm is P-adic Unit | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\Z_p^\times$ be the $p$-adic units.
Let $a \in \Q_p$.
Then there exists $n \in \Z$ such that:
:$p^n a \in \Z_p^\times$
where
:$p^n = \norm a_p$ | From P-adic Norm of p-adic Number is Power of p, there exists $v \in \Z$ such that $\norm a_p = p^{-v}$.
Let $n = -v$.
Now:
{{begin-eqn}}
{{eqn | l = \norm{p^n a}_p
| r = \norm{p^n }_p \norm a_p
| c = Norm axiom (N2) (Multiplicativity)
}}
{{eqn | r = p^{-n} \norm a_p
| c = {{Defof|P-adic Norm|$p$-adic... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\Z_p^\times$ be the [[Definition:P-adic Unit|$p$-adic units]].
Let $a \in \Q_p$.
Then there exists $n \in \Z$ such that:
:$p^n a \in \Z_p^\tim... | From [[P-adic Norm of p-adic Number is Power of p]], there exists $v \in \Z$ such that $\norm a_p = p^{-v}$.
Let $n = -v$.
Now:
{{begin-eqn}}
{{eqn | l = \norm{p^n a}_p
| r = \norm{p^n }_p \norm a_p
| c = [[Definition:Norm on Division Ring|Norm axiom (N2) (Multiplicativity)]]
}}
{{eqn | r = p^{-n} \norm a... | P-adic Number times P-adic Norm is P-adic Unit | https://proofwiki.org/wiki/P-adic_Number_times_P-adic_Norm_is_P-adic_Unit | https://proofwiki.org/wiki/P-adic_Number_times_P-adic_Norm_is_P-adic_Unit | [
"P-adic Number Theory",
"P-adic Units"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:P-adic Unit"
] | [
"P-adic Norm of p-adic Number is Power of p",
"Definition:Norm/Division Ring",
"Definition:Rational Number",
"P-adic Unit has Norm Equal to One"
] |
proofwiki-16690 | Solution to Simultaneous Homogeneous Linear First Order ODEs with Constant Coefficients | Consider the system of linear first order ordinary differential equations with constant coefficients:
{{begin-eqn}}
{{eqn | n = 1
| l = \dfrac {\d y} {\d x} + a y + b z
| r = 0
}}
{{eqn | n = 2
| l = \dfrac {\d x} {\d z} + c y + d z
| r = 0
}}
{{end-eqn}}
The general solution to $(1)$ and $(2)$ ... | We look for solutions to $(1)$ and $(2)$ of the form:
{{begin-eqn}}
{{eqn | n = 3
| l = y
| r = A e^{k x}
}}
{{eqn | n = 4
| l = z
| r = B e^{k x}
}}
{{end-eqn}}
We do of course have the Trivial Solution of Homogeneous Linear 1st Order ODE:
:$y = z = 0$
which happens when $A = B = 0$.
So let us ... | Consider the [[Definition:System of Differential Equations|system]] of [[Definition:Linear First Order ODE with Constant Coefficients|linear first order ordinary differential equations with constant coefficients]]:
{{begin-eqn}}
{{eqn | n = 1
| l = \dfrac {\d y} {\d x} + a y + b z
| r = 0
}}
{{eqn | n = 2
... | We look for [[Definition:Solution to Differential Equation|solutions]] to $(1)$ and $(2)$ of the form:
{{begin-eqn}}
{{eqn | n = 3
| l = y
| r = A e^{k x}
}}
{{eqn | n = 4
| l = z
| r = B e^{k x}
}}
{{end-eqn}}
We do of course have the [[Trivial Solution of Homogeneous Linear 1st Order ODE]]:
... | Solution to Simultaneous Homogeneous Linear First Order ODEs with Constant Coefficients | https://proofwiki.org/wiki/Solution_to_Simultaneous_Homogeneous_Linear_First_Order_ODEs_with_Constant_Coefficients | https://proofwiki.org/wiki/Solution_to_Simultaneous_Homogeneous_Linear_First_Order_ODEs_with_Constant_Coefficients | [
"Systems of Differential Equations",
"Linear First Order ODEs"
] | [
"Definition:Differential Equation/System",
"Definition:Linear First Order Ordinary Differential Equation/Constant Coefficients",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Linear Combination",
"Definition:Root of Polynomial",
"Definition:Quadratic Equation"
] | [
"Definition:Differential Equation/Solution",
"Trivial Solution of Homogeneous Linear 1st Order ODE",
"Definition:Differential Equation/Solution",
"Definition:Zero (Number)",
"Definition:Root of Polynomial",
"Definition:Quadratic Equation",
"Definition:Determinant",
"Definition:Distinct",
"Definition... |
proofwiki-16691 | Trivial Solution of Homogeneous Linear 1st Order ODE | The homogeneous linear first order ODE:
:$\dfrac {\d y} {\d x} + \map Q x y = 0$
has the particular solution:
:$\map y x = 0$
that is, the zero constant function.
This particular solution is referred to as the '''trivial solution'''. | We have:
:$\map {\dfrac {\d} {\d x} } 0 = 0$
from which:
:$\dfrac {\d y} {\d x} + \map Q x y = 0$
Hence the result.
{{Qed}}
Category:Linear First Order ODEs
lcnebp3texs5ohrqzvn56sr3sek9gsg | The [[Definition:Homogeneous Differential Equation|homogeneous]] [[Definition:Linear First Order Ordinary Differential Equation|linear first order ODE]]:
:$\dfrac {\d y} {\d x} + \map Q x y = 0$
has the [[Definition:Particular Solution to Differential Equation|particular solution]]:
:$\map y x = 0$
that is, the [[Def... | We have:
:$\map {\dfrac {\d} {\d x} } 0 = 0$
from which:
:$\dfrac {\d y} {\d x} + \map Q x y = 0$
Hence the result.
{{Qed}}
[[Category:Linear First Order ODEs]]
lcnebp3texs5ohrqzvn56sr3sek9gsg | Trivial Solution of Homogeneous Linear 1st Order ODE | https://proofwiki.org/wiki/Trivial_Solution_of_Homogeneous_Linear_1st_Order_ODE | https://proofwiki.org/wiki/Trivial_Solution_of_Homogeneous_Linear_1st_Order_ODE | [
"Linear First Order ODEs"
] | [
"Definition:Homogeneous Differential Equation",
"Definition:Linear First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Constant Mapping",
"Definition:Differential Equation/Solution/Particular Solution"
] | [
"Category:Linear First Order ODEs"
] |
proofwiki-16692 | Simultaneous Homogeneous Linear First Order ODEs/Examples/y' - 3y + 2z = 0, y' + 4y - z = 0 | Consider the system of linear first order ordinary differential equations with constant coefficients:
{{begin-eqn}}
{{eqn | n = 1
| l = \dfrac {\d y} {\d x} - 3 y + 2 z
| r = 0
}}
{{eqn | n = 2
| l = \dfrac {\d x} {\d z} + 4 y - z
| r = 0
}}
{{end-eqn}}
The general solution to $(1)$ and $(2)$ co... | Using the technique of Solution to Simultaneous Homogeneous Linear First Order ODEs with Constant Coefficients, we calculate the roots of the quadratic equation:
:$\paren {k + a} \paren {k + d} - b c = 0$
where:
{{begin-eqn}}
{{eqn | l = a
| r = -3
}}
{{eqn | l = b
| r = 2
}}
{{eqn | l = c
| r = 4
}}
... | Consider the [[Definition:System of Differential Equations|system]] of [[Definition:Linear First Order ODE with Constant Coefficients|linear first order ordinary differential equations with constant coefficients]]:
{{begin-eqn}}
{{eqn | n = 1
| l = \dfrac {\d y} {\d x} - 3 y + 2 z
| r = 0
}}
{{eqn | n = 2
... | Using the technique of [[Solution to Simultaneous Homogeneous Linear First Order ODEs with Constant Coefficients]], we calculate the [[Definition:Root of Polynomial|roots]] of the [[Definition:Quadratic Equation|quadratic equation]]:
:$\paren {k + a} \paren {k + d} - b c = 0$
where:
{{begin-eqn}}
{{eqn | l = a
... | Simultaneous Homogeneous Linear First Order ODEs/Examples/y' - 3y + 2z = 0, y' + 4y - z = 0 | https://proofwiki.org/wiki/Simultaneous_Homogeneous_Linear_First_Order_ODEs/Examples/y'_-_3y_+_2z_=_0,_y'_+_4y_-_z_=_0 | https://proofwiki.org/wiki/Simultaneous_Homogeneous_Linear_First_Order_ODEs/Examples/y'_-_3y_+_2z_=_0,_y'_+_4y_-_z_=_0 | [
"Examples of Systems of Differential Equations",
"Examples of Linear First Order ODEs"
] | [
"Definition:Differential Equation/System",
"Definition:Linear First Order Ordinary Differential Equation/Constant Coefficients",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Linear Combination"
] | [
"Solution to Simultaneous Homogeneous Linear First Order ODEs with Constant Coefficients",
"Definition:Root of Polynomial",
"Definition:Quadratic Equation",
"Definition:Root of Polynomial",
"Definition:Differential Equation/Solution",
"Definition:Differential Equation/Solution/General Solution"
] |
proofwiki-16693 | Linear First Order ODE/y' - y = x^2 | The linear first order ODE:
:$(1): \quad \dfrac {\d y} {\d x} - y = x^2$
has the general solution:
:$y = C e^x - \paren {x^2 + 2 x + 2}$ | $(1)$ is a linear first order ODE in the form:
:$\dfrac {\d y} {\d x} + \map P x y = \map Q x$
where:
:$\map P x = -1$
:$\map Q x = x^2$
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 ... | The [[Definition:Linear First Order ODE|linear first order ODE]]:
:$(1): \quad \dfrac {\d y} {\d x} - y = x^2$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$y = C e^x - \paren {x^2 + 2 x + 2}$ | $(1)$ is a [[Definition:Linear First Order ODE|linear first order ODE]] in the form:
:$\dfrac {\d y} {\d x} + \map P x y = \map Q x$
where:
:$\map P x = -1$
:$\map Q x = x^2$
Thus:
{{begin-eqn}}
{{eqn | l = \int \map P x \rd x
| r = -\int \rd x
| c =
}}
{{eqn | r = -x
| c =
}}
{{eqn | ll= \leadsto... | Linear First Order ODE/y' - y = x^2 | https://proofwiki.org/wiki/Linear_First_Order_ODE/y'_-_y_=_x^2 | https://proofwiki.org/wiki/Linear_First_Order_ODE/y'_-_y_=_x^2 | [
"Examples of Linear First Order ODEs"
] | [
"Definition:Linear First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Linear First Order Ordinary Differential Equation",
"Solution to Linear First Order Ordinary Differential Equation/Solution by Integrating Factor",
"Primitive of x squared by Exponential of a x"
] |
proofwiki-16694 | Linear First Order ODE/y' - y = e^x/y(0) = 0 | Consider the linear first order ODE:
:$(1): \quad \dfrac {\d y} {\d x} - y = e^x$
subject to the initial condition:
:$\map y 0 = 0$
$(1)$ has the particular solution:
:$y = x e^x$ | $(1)$ is a linear first order ODE in the form:
:$\dfrac {\d y} {\d x} + \map P x y = \map Q x$
where:
:$\map P x = -1$
:$\map Q x = e^x$
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 ... | Consider the [[Definition:Linear First Order ODE|linear first order ODE]]:
:$(1): \quad \dfrac {\d y} {\d x} - y = e^x$
subject to the [[Definition:Initial Condition|initial condition]]:
:$\map y 0 = 0$
$(1)$ has the [[Definition:Particular Solution of Differential Equation|particular solution]]:
:$y = x e^x$ | $(1)$ is a [[Definition:Linear First Order ODE|linear first order ODE]] in the form:
:$\dfrac {\d y} {\d x} + \map P x y = \map Q x$
where:
:$\map P x = -1$
:$\map Q x = e^x$
Thus:
{{begin-eqn}}
{{eqn | l = \int \map P x \rd x
| r = -\int \rd x
| c =
}}
{{eqn | r = -x
| c =
}}
{{eqn | ll= \leadsto... | Linear First Order ODE/y' - y = e^x/y(0) = 0 | https://proofwiki.org/wiki/Linear_First_Order_ODE/y'_-_y_=_e^x/y(0)_=_0 | https://proofwiki.org/wiki/Linear_First_Order_ODE/y'_-_y_=_e^x/y(0)_=_0 | [
"Examples of Linear First Order ODEs"
] | [
"Definition:Linear First Order Ordinary Differential Equation",
"Definition:Initial Condition",
"Definition:Differential Equation/Solution/Particular Solution"
] | [
"Definition:Linear First Order Ordinary Differential Equation",
"Solution to Linear First Order Ordinary Differential Equation/Solution by Integrating Factor",
"Primitive of Constant",
"Definition:Initial Condition",
"Definition:Differential Equation/Solution/Particular Solution"
] |
proofwiki-16695 | Linear Second Order ODE/y'' + 4 y' + 5 y = 0 | The second order ODE:
:$(1): \quad y' ' + 4 y' + 5 y = 0$
has the general solution:
:$y = e^{-2 x} \paren {C_1 \cos x + C_2 \sin x}$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad m^2 + 4 m + 5 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = -2 + i$
:$m_2 = -2 - i$
So from Solution of Constant Coefficient Homogeneous LSO... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' + 4 y' + 5 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = e^{-2 x} \paren {C_1 \cos x + C_2 \sin x}$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad m^2 + 4 m + 5 = 0$
From [[Solution to Quadratic Equation with Real Coefficients]], ... | Linear Second Order ODE/y'' + 4 y' + 5 y = 0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_4_y'_+_5_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_4_y'_+_5_y_=_0 | [
"Examples of Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential Equation/Solution/General Solut... |
proofwiki-16696 | Linear Second Order ODE/y'' + 4 y' + 5 y = 2 exp -2 x | The second order ODE:
:$(1): \quad y'' + 4 y' + 5 y = 2 e^{-2 x}$
has the general solution:
:$y = e^{-2 x} \paren {C_1 \cos x + C_2 \sin x + 2}$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 4$
:$q = 5$
:$\map R x = 2 e^{-2 x}$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$y'' + 4 y' + 5 y = 0$
From Linear Second Or... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' + 4 y' + 5 y = 2 e^{-2 x}$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = e^{-2 x} \paren {C_1 \cos x + C_2 \sin x + 2}$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 4$
:$q = 5$
:$\map R x = 2 e^{-2 x}$
First we establish the solution of the corresponding [[Definition:Constant Coefficient Homogeneous L... | Linear Second Order ODE/y'' + 4 y' + 5 y = 2 exp -2 x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_4_y'_+_5_y_=_2_exp_-2_x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_4_y'_+_5_y_=_2_exp_-2_x | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Undetermined Coefficients"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' + 4 y' + 5 y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Differential Equation/Solution/Particular Solution",
"... |
proofwiki-16697 | Linear Second Order ODE/y'' + y = exp -x cos x | The second order ODE:
:$(1): \quad y'' + y = e^{-x} \cos x$
has the general solution:
:$y = \dfrac {e^{-x} } 5 \paren {\cos x - 2 \sin x} + C_1 \sin x + C_2 \cos x$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 0$
:$q = 1$
:$\map R x = e^{-x} \cos x$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$y'' + y = 0$
From Linear Second Order OD... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' + y = e^{-x} \cos x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = \dfrac {e^{-x} } 5 \paren {\cos x - 2 \sin x} + C_1 \sin x + C_2 \cos x$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 0$
:$q = 1$
:$\map R x = e^{-x} \cos x$
First we establish the solution of the corresponding [[Definition:Constant Coefficient Homogeneou... | Linear Second Order ODE/y'' + y = exp -x cos x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_y_=_exp_-x_cos_x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_y_=_exp_-x_cos_x | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Undetermined Coefficients"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' + y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Differential Equation/Solution/Particular Solution",
"Method of... |
proofwiki-16698 | Linear Second Order ODE/y'' - 5 y' + 6 y = 0 | The linear second order ODE:
:$(1): \quad y' ' - 5 y' + 6 y = 0$
has the general solution:
:$y = C_1 e^{2 x} + C_2 e^{3 x}$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad: m^2 - 5 m + 6 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = 2$
:$m_2 = 3$
These are real and unequal.
So from Solution of Constant Coeffici... | The [[Definition:Linear Second Order ODE|linear second order ODE]]:
:$(1): \quad y' ' - 5 y' + 6 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^{2 x} + C_2 e^{3 x}$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad: m^2 - 5 m + 6 = 0$
From [[Solution to Quadratic Equation with Real Coefficients]]... | Linear Second Order ODE/y'' - 5 y' + 6 y = 0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_5_y'_+_6_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_5_y'_+_6_y_=_0 | [
"Examples of Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Linear Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Definition:Real Number",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential Equ... |
proofwiki-16699 | Linear Second Order ODE/y'' - 5 y' + 6 y = cos x + sin x | The second order ODE:
:$(1): \quad y' ' - 5 y' + 6 y = \cos x + \sin x$
has the general solution:
:$y = C_1 e^{2 x} + C_2 e^{3 x} + \dfrac {\cos x} 5$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE in the form:
:$y' ' + p y' + q y = \map R x $
where:
:$p = -5$
:$q = 5$
:$\map R x = \cos x + \sin x$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$y' ' - 5 y' + 6 y = 0$
From Linear ... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' - 5 y' + 6 y = \cos x + \sin x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^{2 x} + C_2 e^{3 x} + \dfrac {\cos x} 5$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] in the form:
:$y' ' + p y' + q y = \map R x $
where:
:$p = -5$
:$q = 5$
:$\map R x = \cos x + \sin x$
First we establish the solution of the corresponding [[Definition:Constant Coefficient Homog... | Linear Second Order ODE/y'' - 5 y' + 6 y = cos x + sin x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_5_y'_+_6_y_=_cos_x_+_sin_x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_5_y'_+_6_y_=_cos_x_+_sin_x | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Undetermined Coefficients"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' - 5 y' + 6 y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Differential Equation/Solution/Particular Solution",
"... |
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