hcju/proofwikips huggingface dataset:
Equation of Cardioid/Polar
Let $C$ be a cardioid embedded in a polar coordinate plane such that: :its stator 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: :$r = 2 a \paren {1 + \cos \theta}$
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 $16 a$.
Arc Length for Polar Curve
Let $a$ and $b$ be real numbers. Let $\mathcal C$ be a simple curve continuous on $\closedint a b$ and continuously differentiable on $\openint a b$. Let $\mathcal C$ be described by the parametric equations: :$\begin {cases} x & = r \cos \theta \\ y & = r \sin \theta \end {cases}$ where: :$r$ is a function of $\theta$ :$\theta \in \closedint a b$. Then the length $s$ of $\mathcal C$ is given by: :$\displaystyle s = \int_a^b \sqrt {r^2 + \paren {\frac {\d r} {\d \theta} }^2} \rd \theta$
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.
Number of Petals of Even Index Rhodonea Curve
Let $n$ be an even (strictly) 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 $2 n$ petals.
Equation of Tractrix
Let $S$ be a cord of length $a$ situated as a (straight) line segment whose endpoints are $P$ and $T$. Let $S$ be aligned along the $x$-axis of a cartesian plane with $T$ at the origin and $P$ therefore at the point $\tuple {a, 0}$. Let $T$ be dragged along the $y$-axis. :File:Tractrix.png
Equation of Witch of Agnesi/Parametric Form
The equation of the Witch of Agnesi can be presented in paremetric form as: :$\begin {cases} x = 2 a \cot \theta \\ y = a \paren {1 - \cos 2 \theta} \end {cases}$
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: :$\AA = \dfrac {3 a^2} 2$
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 $F$.
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 quadrant is generated  :For $0 \le t$, the section in the $1$st quadrant is generated.
Evolute of Ellipse/Cartesian Form
The evolute of $E$ is given by the Cartesian equation: :$\paren {a x}^{2 / 3} + \paren {b y}^{2 / 3} = \paren {a^2 - b^2}^{2 / 3}$
Evolute of Ellipse/Parametric Form
The evolute of $E$ can be expressed using the parametric equation: :$\begin {cases} a x = \paren {a^2 - b^2} \cos^3 \theta \\ b y = \paren {a^2 - b^2} \sin^3 \theta \end {cases}$
Evolute of Ellipse
Let $E$ be an ellipse embedded in a Cartesian plane with the equation: :$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$
Equation of Ovals of Cassini
Let $P_1$ and $P_2$ be points in the plane such that $P_1 P_2 = 2 a$ for some constant $a$. Let $b$ be a real constant.
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'''.
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'''.
Lemniscate of Bernoulli is Special Case of Ovals of Cassini
The lemniscate of Bernoulli is a special case of the ovals of Cassini.
Cramer's Rule
Let $n \in \N$. Let $b_1, b_2, \dots, b_n$ be real numbers. Let $\mathbf b = \tuple {b_1, b_2, \dots, b_n}^T$. Let $x_1, x_2, \dots, x_n$ be real numbers. Let $\mathbf x = \tuple {x_1, x_2, \dots, x_n}^T$. Let $A$ be an invertible $n \times n$ matrix with coefficients in $\R$. For each $i \in \set {1, \dots, n}$, let $A_i$ be the matrix obtained by replacing the $i$th column with $\mathbf b$. Let: :$A \mathbf x = \mathbf b$ Then: :$\mathbf x_i = \dfrac {\map \det {A_i} } {\map \det A}$ for each $i \in \set {1, \dots, n}$.
Equation of Limaçon of Pascal
Let $C$ be a circle of diameter $a$ whose circumference passes through the origin $O$ and whose diameter through $O$ lies on the horizontal. Let $b$ be a real constant.
Equation of Limaçon of Pascal/Polar Form
The limaçon of Pascal can be defined by the polar equation: :$r = b + a \cos \theta$
Equation of Cissoid of Diocles
Let $C$ be a circle of radius $a$ whose circumference passes through the origin $O$ and whose diameter through $O$ lies on the horizontal.
Equation of Cissoid of Diocles/Polar Form
The cissoid of Diocles can be defined by the polar equation: :$r = 2 a \sin \theta \tan \theta$
Equation of Cissoid of Diocles/Parametric Form
The cissoid of Diocles can be defined by the parametric equation: :$\begin {cases} x = 2 a \sin^2 \theta \\ y = \dfrac {2 a \sin^3 \theta} {\cos \theta} \end {cases}$
Primitive of One plus x Squared over One plus Fourth Power of x
:$\displaystyle \int \frac {x^2 + 1} {x^4 + 1} \rd x = \frac 1 {\sqrt 2} \map \arctan {\frac 1 {\sqrt 2} \paren {x - \frac 1 x} } + C$
Primitive of Minus One plus x Squared over One plus Fourth Power of x
:$\displaystyle \int \frac {x^2 - 1} {x^4 + 1} \rd x = \frac 1 {2 \sqrt 2} \ln \size {\frac {x^2 - \sqrt 2 x + 1} {x^2 + \sqrt 2 x + 1} } + C$
Dissection of Square into 8 Acute Triangles
A square can be dissected into $8$ acute triangles.
Dissection of Square into 9 Acute Triangles
A square can be dissected into $9$ acute triangles.
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 $C_1$.
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: :$\displaystyle \map {\pr_1} w = \bigcup \bigcap w$ where $\displaystyle \bigcup$ and $\displaystyle \bigcap$ denote union and intersection respectively.
Intersection of Doubleton
Let $\set {x, y}$ be a doubleton. Then: :$\displaystyle \bigcap \set {x, y} = x \cup y$
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: :$\displaystyle \map {\pr_2} w = \begin {cases} \displaystyle \map \bigcup {\bigcup w \setminus \bigcap w} & : \displaystyle \bigcup w \ne \bigcap w \\ \displaystyle \bigcup \bigcup w & : \displaystyle \bigcup w = \bigcap w \end {cases}$ where: :$\displaystyle \bigcup$ and $\displaystyle \bigcap$ denote union and intersection respectively. :$\setminus$ denotes the set difference operator.
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 mapping $f: S_1 \to S_2$ such that $f$ is an order isomorphism.
Pi as Sum of Sequence of Reciprocal of Product of Three Consecutive Integers/Lemma
:$\displaystyle \iiint \dfrac x {x^2 + 1} \rd x \rd x \rd x = x \map \arctan x + \dfrac {\paren {x^2 - 1} \map \ln {x^2 + 1} - 3 x^2} 4$ with all integration constants at $0$.
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 $\set {1, 2, \ldots, n}$ : :$S$ has a supremum and, in either case: :$\sup S = \max \set {\sup S_1, \sup S_2, \ldots, \sup S_n}$
Equivalence Class in P-adic Integers Contains Unique Coherent Sequence/Lemma 1
:$\forall j \in \N: \norm {\beta_{\map N {j + 1} } - \beta_{\map N j} }_p \le p^{-\paren {j + 1} }$
Equivalence Class in P-adic Integers Contains Unique Coherent Sequence/Lemma 2
:$\forall j \in \N: \norm {\alpha_{j + 1} - \alpha_j }_p \le p^{-\paren {j + 1} }$
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: :$x y \paren {\paren {y'}^2 - 1} + \paren {x^2 - y^2 - a^2 + b^2} y' = 0$
Equivalence Class in P-adic Integers Contains Unique Coherent Sequence/Lemma 3
:$\sequence {\alpha_n}$ and $\sequence {\beta_n}$ are representatives of the same equivalence class in $\Q_p$.
Equivalence Class in P-adic Integers Contains Unique Coherent Sequence/Lemma 4
:$\sequence {\alpha_j}$ is the only coherent sequence that represents $\mathbf a$.
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 successive partial differentiation of $(1)$.
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 as a quotient of Cauchy sequences. Let $\sequence{\alpha_n}$ and $\sequence{\beta_n}$ be Cauchy sequences in $\struct {Q, \norm {\,\cdot\,}_p}$. Then:  :$\sequence {\alpha_n}$ and $\sequence {\beta_n}$ are representatives of the same equivalence class in $\Q_p$ : :the sequence $\sequence {\alpha_n - \beta_n}$ is a null sequence.
Inverse of Vandermonde Matrix/Eisinberg Formula
Let: {{begin-eqn}} {{eqn | l = \prod_{k \mathop = 1}^n \paren {x - x_k}       | r = a_nx^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 =  Viete's Formulas and {{Defof|Elementary Symmetric Function}} }} {{eqn | l = W_n        | r =  \begin{bmatrix}   1         &  x_1      & \cdots & x_1^{n-1} \\   1         & x_2       & \cdots & x_2^{n-1} \\ \vdots      & \vdots    & \ddots & \vdots    \\   1         & x_1^{n-1} & \cdots & x_n^{n-1} \\ \end{bmatrix}       | c = {{Defof|Vandermonde Matrix}} of Order $n$ }} {{end-eqn}} Let $W_n$ have a matrix inverse $W_n^{-1} = \begin {bmatrix} d_{ij} \end {bmatrix}$.  Let $a_n = \map {e_0} {x_1, \ldots, x_n} = 1$. Then: {{begin-eqn}} {{eqn | n = 1       | l = d_{ij}       | r = \dfrac {\displaystyle \sum_{k \mathop = 0}^{n - i} a_{i + k} \, x_j^k} {\displaystyle \prod_{m \mathop = 1, m \mathop \ne j }^n \paren {x_j - x_m} }       | c = for $i, j = 1, \ldots, n$ }} {{eqn | r = \dfrac {\displaystyle \sum_{k \mathop = 0}^{n - i} \paren {-1}^{n - i - k} \map {e_{n - i - k} } {x_1, \ldots, x_n} \, x_j^k} {\displaystyle \prod_{m \mathop = 1, m \mathop \ne j }^n \paren {x_j - x_m} }       | c = for $i, j = 1, \ldots, n$ }} {{end-eqn}}
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: :$y = \displaystyle \int \map f x \rd x + C$ where $\displaystyle \int \map f x \rd x$ denotes the primitive of $f$.
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: :$y = y_0 + \displaystyle \int_{x_0}^x \map f \xi \rd \xi$ where $\displaystyle \int \map f x \rd x$ denotes the primitive of $f$.
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: :$y = \dfrac 1 {10} \paren {3 \sin x - \cos x} + C e^{3 x}$
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: :$\ds y = e^{-a x} \paren {\int e^{a x} \map Q x \rd x + C}$
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: :$\displaystyle y = e^{-a x} \int_{x_0}^x e^{a \xi} \map Q \xi \rd \xi + y_0 e^{a \paren {x - x_0} }$
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: :$y = \displaystyle e^{-x} \int_1^x \dfrac {e^\xi \rd \xi} \xi$
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 arbitrary constant is $0$ plus: :the complementary function to $(1)$.
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.
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: :$y = e^{-x} \paren {A \cos x + B \sin x}$
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: :$y'' + 2 y' + 2 y = 0$
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: :$y = C_1 \, \map \exp {\paren {\dfrac 7 2 + \dfrac {\sqrt {69} } 2} x} + C_2 \, \map \exp {\paren {\dfrac 7 2 - \dfrac {\sqrt {69} } 2} x} + \dfrac 1 {625} \paren {-125 x^3 + 525 x^2 - 1620 x + 2603}$
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: :$y = C_1 \, \map \exp {\paren {\dfrac 7 2 + \dfrac {\sqrt {69} } 2} x} + C_2 \, \map \exp {\paren {\dfrac 7 2 - \dfrac {\sqrt {69} } 2} x}$
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.
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: :$y = C_1 \, \map \exp {\paren {1 + \sqrt 6} x} + C_2 \, \map \exp {\paren {1 - \sqrt 6} x}$
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: :$y = C_1 \, \map \exp {\paren {1 + \sqrt 6} x} + C_2 \, \map \exp {\paren {1 - \sqrt 6} x} + \dfrac 1 {116} \paren {\sin 3 x - 17 \cos 3 x}$
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: :$y_p = \dfrac 1 {116} \paren {\sin 3 x - 17 \cos 3 x}$
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: :$y = C_1 \sin k x + C_2 \cos k x - \dfrac 3 4 x \cos 2 x$
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.
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: :$y = C_1 e^{5 x} + C_2 e^{-x}$
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: :$y = \dfrac {e^{5 x} } {375} + \dfrac {4 e^{-x} } 3 - \dfrac {x^2} 5 + \dfrac {8 x} {25} - \dfrac {42} {125}$
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  $\norm x_p = 1$
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 :$p^n = \norm a_p$
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 $(2)$ consists of the linear combinations of the following: {{begin-eqn}} {{eqn | l = y       | r = A_1 e^{k_1 x} }} {{eqn | l = z       | r = B_1 e^{k_1 x} }} {{end-eqn}} and: {{begin-eqn}} {{eqn | l = y       | r = A_2 e^{k_2 x} }} {{eqn | l = z       | r = B_2 e^{k_2 x} }} {{end-eqn}} where $A_1 : B_1 = A_2 : B_2 = r$ where $r$ is either of the roots of the quadratic equation: :$\paren {k + a} \paren {k + d} - b c = 0$
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'''.
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 $(2)$ consists of the linear combinations of the following: {{begin-eqn}} {{eqn | l = y       | r = C_1 e^{5 x} + C_2 e^{-x} }} {{eqn | l = z       | r = -C_1 e^{5 x} + 2 C_2 e^{-x} }} {{end-eqn}}
Particular Solution of Constant Coefficient Linear nth Order ODE
Consider the linear $n$th order ODE with constant coefficients: :$(1): \quad \displaystyle \sum_{k \mathop = 0}^n a_k \dfrac {\d^k y} {d x^k} = \map R x$ Let $(1)$ have the following $n$ initial conditions: :$(2): \quad y = y_0, \dfrac {\d y} {\d x} = y_1, \dotsc, \dfrac {\d^{n - 1} y} {\d x^{n - 1} } = y_{n - 1}$ when $x = x_0$. Then there exists exactly one particular solution of $(1)$ which satisfies $(2)$.
Particular Solution of System of Constant Coefficient Linear 1st Order ODEs
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}} Let $(1)$ and $(2)$ have the following $n$ initial conditions: :$(3): \quad y = y_0, z = z_0$ when $x = x_0$. Then there exists exactly one particular solution of $(1)$ and $(2)$ which satisfies $(3)$.
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: :$y = x e^x$
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: :$y = e^{-2 x} \paren {C_1 \cos x + C_2 \sin x}$
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: :$y = C_1 e^{2 x} + C_2 e^{3 x}$
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: :$y = C_1 e^{2 x} + C_2 e^{3 x} + \dfrac {\cos x} 5$
Lindelöf's Lemma/Lemma
Let $C$ be a set of open real sets. Then there is a countable subset $D$ of $C$ such that: :$\displaystyle \bigcup_{O \mathop \in D} O = \bigcup_{O \mathop \in C} O$
Linear Second Order ODE/y'' - 4 y = x^2 - 3 x - 4
The second order ODE: :$(1): \quad y'' - 4 y = x^2 - 3 x - 4$ has the general solution: :$y = C_1 e^{2 x} + C_2 e^{-2 x} - \dfrac {x^2} 4 + \dfrac {3 x} 4 + \dfrac 7 8$
Linear Second Order ODE/y'' + 2 y' + y = x exp -x
The second order ODE: :$(1): \quad y'' + 2 y' + y = x e^{-x}$ has the general solution: :$y = e^{-x} \paren {C_1 + C_2 x + \dfrac {x^3} 6}$
Angular Momentum Commutation Rules
Let $J_x$, $J_y$ and $J_z$ denote the angular momentum operators. Then: {{begin-eqn}} {{eqn | l = \sqbrk {J_x, J_y}       | r = i J_z }} {{eqn | l = \sqbrk {J_y, J_z}       | r = i J_x }} {{eqn | l = \sqbrk {J_z, J_x}       | r = i J_y }} {{end-eqn}} where $\sqbrk {\, \cdot, \cdot \,}$ denotes the commutator operator.
P-adic Norm of P-adic Expansion is determined by First Nonzero Coefficient
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers as a quotient of Cauchy sequences. Let $\displaystyle \sum_{i \mathop = m}^\infty d_i p^i$ be a $p$-adic expansion. Let $\mathbf a$ be the equivalence class in $\Q_p$ containing $\displaystyle \sum_{i \mathop = m}^\infty d_i p^i$. Let $l$ be the index of the first non-zero coefficient in the $p$-adic expansion: :$l = \min \set {i: i \ge m \land d_i \ne 0}$ Then: :$\norm {\mathbf a}_p = p^{-l}$
Eventually Constant Sequence Converges to Constant
:$\displaystyle \lim_{n \mathop \to \infty} x_n = \lambda$
Non-Zero Complex Numbers under Multiplication form Group
Let $\C_{\ne 0}$ be the set of complex numbers without zero, that is: :$\C_{\ne 0} = \C \setminus \set 0$ The structure $\struct {\C_{\ne 0}, \times}$ is a group.
Real Numbers under Addition form Group
Let $\R$ be the set of real numbers. The structure $\struct {\R, +}$ is an infinite abelian group.
General Linear Group is not Abelian
Let $K$ be a field whose zero is $0_K$ and unity is $1_K$. Let $\GL {n, K}$ be the general linear group of order $n$ over $K$. Then $\GL {n, K}$ is not an abelian group.
Group of Unitary Matrices under Multiplication is not Abelian
Let $n > 1$ be a natural number. Then the group of unitary matrices $\map U n$ is not abelian.
Closure in Subspace/Corollary 1
Let $W \subseteq S$ and let $\map {\cl_T} W$ denote the closure of $W$ in $T$. Let $\map {\cl_H} {W \cap H}$ denote the closure of $W \cap H$ in $T_H$. Then: :$\map {\cl_H} {W \cap H} \subseteq \map {\cl_T} W \cap H$
Complex Numbers form Vector Space over Themselves
The set of complex numbers $\C$, with the operations of addition and multiplication, forms a vector space.
Quaternions form Vector Space over Themselves
The set of quaternions $\H$, with the operations of addition and multiplication, forms a vector space.