hcju/proofwikips huggingface dataset:
Length of Arch of Sine Function
The length of one arch of the sine function: :$y = \sin x$ is given by: :$L = 2 \sqrt 2 \map E {\dfrac {\sqrt 2} 2}$ where $E$ denotes the incomplete elliptic integral of the second kind.
Length of Lemniscate of Bernoulli
The total length of the lemniscate of Bernoulli given in polar coordinates as: :$r^2 = a^2 \cos 2 \theta$ is given by: {{begin-eqn}} {{eqn | l = L       | r = 4 a \map F {\sqrt 2, \dfrac \pi 4}       | c =  }} {{eqn | r = \dfrac 1 {\sqrt {2 \pi} } \paren {\map \Gamma {\dfrac 1 4} }^2       | c =  }} {{end-eqn}} where $F$ denotes the incomplete elliptic integral of the first kind.
Boundary of Union is Subset of Union of Boundaries
Let $T = \left({S, \tau}\right)$ be a topological space. Let $A, B$ be subsets of $S$. Then: :$\partial \left({A \cup B}\right) \subseteq \partial A \cup \partial B$ where $\partial A$ denotes the boundary of $A$.
Boundary of Intersection is Subset of Union of Boundaries
Let $T = \struct {S, \tau}$ be a topological space. Let $A, B$ be subsets of $S$. Then: :$\partial \paren {A \cap B} \subseteq \partial A \cup \partial B$ where $\partial A$ denotes the boundary of $A$.
Complete Elliptic Integral of the First Kind as Power Series
The '''complete elliptic integral of the first kind''': :$\displaystyle \map K k = \int_0^{\pi / 2} \frac {\rd \phi} {\sqrt {1 - k^2 \sin^2 \phi} } = \int_0^1 \frac {\rd v} {\sqrt {\paren {1 - v^2} \paren {1 - k^2 v^2} } }$ can be expressed as the power series: {{begin-eqn}} {{eqn | l = \map K k       | r = \frac \pi 2 \sum_{i \mathop \ge 0} \paren {\prod_{j \mathop = 1}^i \frac {2 j - 1} {2 j} }^2 k^{2 i}       | c =  }} {{eqn | r = \frac \pi 2 \paren {1 + \paren {\frac 1 2}^2 k^2 + \paren {\frac {1 \cdot 3} {2 \cdot 4} }^2 k^4 + \paren {\frac {1 \cdot 3 \cdot 5} {2 \cdot 4 \cdot 6} }^2 k^6 + \cdots}       | c =  }} {{end-eqn}}
Complete Elliptic Integral of the Second Kind as Power Series
The '''complete elliptic integral of the second kind''': :$\displaystyle \map E k = \int_0^{\pi / 2} \sqrt {1 - k^2 \sin^2 \phi} \, \rd \phi = \int_0^1 \dfrac {\sqrt {1 - k^2 v^2} } {\sqrt {1 - v^2}} \, \rd v$ can be expressed as the power series: {{begin-eqn}} {{eqn | l = \map E k       | r = \frac \pi 2 \sum_{i \mathop \ge 0} \paren {\prod_{j \mathop = 1}^i \frac {2 j - 1} { 2 j} }^2 \frac {k^{2 i} } {1 - 2 i}       | c =  }} {{eqn | r = \frac \pi 2 \paren {1 - \paren {\frac 1 2}^2 k^2 - \paren {\frac {1 \cdot 3} {2 \cdot 4} }^2 \frac {k^4} 3 - \paren {\frac {1 \cdot 3 \cdot 5} {2 \cdot 4 \cdot 6} }^2 \frac {k^6} 5 - \cdots}       | c =  }} {{end-eqn}}
Set is Countable iff Cardinality not greater Aleph Zero
Let $X$ be a set. Then: :$X$ is countable  $\size X \le \aleph_0$ where :$\size X$ denotes the cardinality of $X$, :$\aleph$ denotes the aleph mapping.
Bernoulli's Theorem
Let the probability of the occurrence of an event be $p$. Let $n$ independent trials be made, with $k$ successes. Then: :$\displaystyle \lim_{n \mathop \to \infty} \frac k n = p$
Time of Travel down Brachistochrone
Let a wire $AB$ be curved into the shape of a brachistochrone. Let $AB$ be embedded in a constant and uniform gravitational field where Acceleration Due to Gravity is $g$. Let a bead $P$ be released at $A$ to slide down without friction to $B$. Then the time taken for $P$ to slide from $A$ to $B$ is: :$T = \pi \sqrt {\dfrac a g}$ where $a$ is the radius of the generating circle of the cycloid which forms $AB$.
Time of Travel down Brachistochrone/Corollary
Let a bead $P$ be released from anywhere on the wire between $A$ and $B$ to slide down without friction to $B$. Then the time taken for $P$ to slide to $B$ is: :$T = \pi \sqrt{\dfrac a g}$
Union of Set of Singletons
Let $S$ be a set. Let $T = \set {\set x: x \in S}$ be the set of all singletons of elements of $S$. Then: :$\ds \bigcup T = S$ where $\ds \bigcup T$ denotes the union of $T$.
T1 Space is T1/2 Space
Let $T$ be a $T_1$ topological space. Then $T$ is $T_{\frac 1 2}$ space.
Parabolas Inscribed in Shared Tangent Lines
Let the function $\map f x = A x^2 + B x + C_1$ be a curve embedded in the Euclidean Plane. Let $\map {y_1} x$ be the equation of the tangent line at $\tuple {Q, \map f Q}$ on $f$. Let $\map {y_2} x$ be the equation of the tangent line at $\tuple {-Q, \map f {-Q} }$ on $f$. Then there exists another function $\map g x$ also embedded in the Euclidean Plane defined as: :$\map g x = -A x^2 + B x + C_2$. with: :tangent lines $\map {y_3} x$ being the equation of the tangent line at $\tuple {Q, \map g Q}$ on $g$ and: :$\map {y_4} x$ being the equation of the tangent line at $\tuple {-Q, \map g {-Q} }$ on $g$. so that the tangent lines $y_3$ and $y_4$ inscribe $\map f x$ and the tangent lines $y_1$ and $y_2$ inscribe $\map g x$.
Characterization of T0 Space by Closures of Singletons
Let $T = \left({S, \tau}\right)$ be a topological space. Then :$T$ is a $T_0$ space  ::$\forall x, y \in S: x \ne y \implies x \notin \left\{{y}\right\}^- \lor y \notin \left\{{x}\right\}^-$ where $\left\{{y}\right\}^-$ denotes the closure of $\left\{{y}\right\}$.
Characterization of T0 Space by Distinct Closures of Singletons
Let $T = \left({S, \tau}\right)$ be a topological space. Then :$T$ is a $T_0$ space  ::$\forall x, y \in S: x \ne y \implies \left\{{x}\right\}^- \ne \left\{{y}\right\}^-$ where $\left\{{y}\right\}^-$ denotes the closure of $\left\{{y}\right\}$.
Characterization of T0 Space by Closed Sets
Let $T = \left({S, \tau}\right)$ be a topological space. Then :$T$ is a $T_0$ space  ::for every points $x, y \in S$ if $x \ne y$ then :::there exists a closed subset $F$ of $S$ such that $x \in F$ and $y \notin F$ ::or :::there exists a closed subset $F$ of $S$ such that $x \notin F$ and $y \in F$
Focus of Ellipse from Major and Minor Axis
Let $K$ be an ellipse whose major axis is $2 a$ and whose minor axis is $2 b$. Let $c$ be the distance of the foci of $K$ from the center. Then: :$a^2 = b^2 + c^2$
Equidistance of Ellipse equals Major Axis
Let $K$ be an ellipse whose foci are $F_1$ and $F_2$. Let $P$ be an arbitrary point on $K$. Let $d$ be the constant distance such that: :$d_1 + d_2 = d$ where: : $d_1 = P F_1$ : $d_2 = P F_2$ Then $d$ is equal to the major axis of $K$.
Equidistance of Hyperbola equals Transverse Axis
Let $K$ be an hyperbola whose foci are $F_1$ and $F_2$. Let $P$ be an arbitrary point on $K$. Let $d$ be the constant distance such that: :$\left\lvert{d_1 - d_2}\right\rvert = d$ where: : $d_1 = P F_1$ : $d_2 = P F_2$ Then $d$ is equal to the transverse axis of $K$.
Equivalence of Definitions of Ellipse
The following definitions of an ellipse are equivalent:
Equation of Hyperbola in Reduced Form
Let $K$ be an hyperbola aligned in a cartesian plane in reduced form. Let: :the transverse axis of $K$ have length $2 a$ :the conjugate axis of $K$ have length $2 b$.
Equation of Hyperbola in Reduced Form/Cartesian Frame
The equation of $K$ is: :$\dfrac {x^2} {a^2} - \dfrac {y^2} {b^2} = 1$
Focus of Hyperbola from Transverse and Conjugate Axis
Let $K$ be a hyperbola whose transverse axis is $2 a$ and whose conjugate axis is $2 b$. Let $c$ be the distance of the foci of $K$ from the center. Then: :$c^2 = a^2 + b^2$
Equation of Hyperbola in Reduced Form/Cartesian Frame/Parametric Form
The equation of $K$ in parametric form is: :$x = a \cosh \theta, y = b \sinh \theta$
Set of Condensation Points is Subset of Derivative
Let $T = \left({S, \tau}\right)$ be a topological space. Let $A$ be a subset of $S$. Then: :$A^0 \subseteq A'$ where :$A^0$ denotes the set of condensation points of $A$ :$A'$ denotes the derivative of $A$
Closure of Set of Condensation Points equals Itself
Let $T = \left({S, \tau}\right)$ be a topological space. Let $A$ be a subset of $S$. Then: :$\left({A^0}\right)^- = A^0$ where :$A^0$ denotes the set of condensation points of $A$ :$A^-$ denotes the closure of $A$
First Order ODE/(x + y) dx = (x - y) dy
is a homogeneous differential equation with general solution: :$\arctan \dfrac y x = \ln \sqrt {x^2 + y^2} + C$
Set of Condensation Points is Monotone
Let $T = \left({S, \tau}\right)$ be a topological space. Let $A, B$ be subsets of $S$. Then: :$A \subseteq B \implies {A^0} \subseteq B^0$ where :$A^0$ denotes the set of condensation points of $A$
First Order ODE/x^2 y' = 3 (x^2 + y^2) arctan (y over x) + x y
is a homogeneous differential equation with solution: :$y = x \tan C x^3$
First Order ODE/x sine (y over x) y' = y sine (y over x) + x
is a homogeneous differential equation with solution: :$\cos \dfrac y x + \ln C x = 0$
First Order ODE in form y' = f (a x + b y + c)
The first order ODE: :$\dfrac {\mathrm d y} {\mathrm d x} = f \left({a x + b y + c}\right)$ can be solved by substituting: :$z := a x + b y + c$ to obtain: :$\displaystyle x = \int \frac {\mathrm d z} {b f \left({z}\right) + a}$
First Order ODE/y' = sin^2 (x - y + 1)
The first order ODE: :$\dfrac {\d y} {\d x} = \map {\sin^2} {x - y + 1}^2$ has the general solution: :$\map \tan {x - y + 1} = x + C$
First Order ODE in form y' = F ((a x + b y + c) over (d x + e y + f))
The first order ODE: :$\dfrac {\d y} {\d x} = \map F {\dfrac {a x + b y + c} {d x + e y + f} }$ such that: :$ a e \ne b d$ can be solved by substituting: :$x := z - h$ :$y := w - k$ where: :$h = \dfrac {c e - b f} {a e - b d}$ :$k = \dfrac {a f - c d} {a e - b d}$ to obtain: :$\dfrac {\d w} {\d z} = \map F {\dfrac {a z + b w} {d z + e w} }$ which can be solved by the technique of Solution to Homogeneous Differential Equation.
First Order ODE/(x + y + 4) over (x - y - 6)
The first order ODE: :$(1): \quad \dfrac {\d y} {\d x} = \dfrac {x + y + 4} {x - y - 6}$ has the general solution: :$\map \arctan {\dfrac {y + 5} {x - 1} } = \ln \sqrt {\paren {x - 1}^2 + \paren {y + 5}^2} + C$
First Order ODE/(x + y + 4) over (x + y - 6)
The first order ODE: :$(1): \quad \dfrac {\d y} {\d x} = \dfrac {x + y + 4} {x + y - 6}$ has the general solution: :$y - x = 5 \, \map \ln {x + y - 1} + C$
First Order ODE in form y' = F ((a x + b y + c) over (d x + e y + f)) where a e = b d/Example
=== $\dfrac {\d y} {\d x} = \dfrac {x + y + 4} {x + y - 6}$ === {{:First Order ODE/(x + y + 4) over (x + y - 6)}}
First Order ODE in form y' = F ((a x + b y + c) over (d x + e y + f))/Example
=== $\dfrac {\d y} {\d x} = \dfrac {x + y + 4} {x - y - 6}$ === {{:First Order ODE/(x + y + 4) over (x - y - 6)}}
First Order ODE/(x + (2 over y)) dy + y dx = 0
is an exact differential equation with solution: :$x y + 2 \ln y = C$
First Order ODE/(y - x^3) dx + (x + y^3) dy = 0
is an exact differential equation with solution: :$4 x y - x^4 + y^4 = C$
First Order ODE/(y + y cosine x y) dx + (x + x cosine x y) dy = 0
is an exact differential equation with solution: :$x y + \sin x y = C$
First Order ODE/(sine x sine y - x e^y) dy = (e^y + cosine x cosine y) dx
is an exact differential equation with solution: :$\sin x \cos y + x e^y = C$
First Order ODE/(2 x y^3 + y cosine x) dx + (3 x^2 y^2 + sine x) dy
is an exact differential equation with solution: :$x^2 y^3 + y \sin x = C$
Heart Curve
The following equations define curves that are more or less heart shaped:
Limit Point of Subset is Limit Point of Set
Let $T = \struct {S, \tau}$ be a topological space. Let $A, B$ be subset of $S$ such that :$A \subseteq B$ Let $x$ be a point of $S$. Then: :if $x$ is limit point of $A$, then $x$ is limit point of $B$.
Integrating Factor for First Order ODE/Preliminary Work
Let the first order ordinary differential equation: :$(1): \quad \map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$ be non-homogeneous and not exact. Let $\map \mu x$ be an integrating factor for $(1)$. Let: :$\map P {x, y} := \dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x}$ Then: :$\dfrac 1 \mu = \dfrac {\map P {x, y} } {N \dfrac {\partial \mu} {\partial x} - M  \dfrac {\partial \mu} {\partial y} }$
Integrating Factor for First Order ODE/Function of One Variable
Suppose that: :$g \left({x}\right) = \dfrac {\dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x} } {N \left({x, y}\right)}$ is a function of $x$ only. Then: :$\mu \left({x}\right) = e^{\int g \left({x}\right) \mathrm d x}$ is an integrating factor for $(1)$. Similarly, suppose that: :$h \left({y}\right) = \dfrac {\dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x} } {M \left({x, y}\right)}$ is a function of $y$ only. Then: :$\mu \left({y}\right) = e^{\int -h \left({y}\right) \mathrm d y}$ is an integrating factor for $(1)$.
Integrating Factor for First Order ODE/Function of Sum of Variables
Suppose that: :$g \left({z}\right) = \dfrac {\dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x} } {N \left({x, y}\right) - M \left({x, y}\right)}$ is a function of $z$, where $z = x + y$. Then: :$\mu \left({x + y}\right) = \mu \left({z}\right) = e^{\int g \left({z}\right) \mathrm d z}$ is an integrating factor for $(1)$.
Integrating Factor for First Order ODE/Function of Product of Variables
:$\map g z = \dfrac {\dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x}} {N y - M x}$ is a function of $z$, where $z = x y$. Then: :$\map \mu {x y} = \map \mu z = e^{\int \map g z \d z}$ is an integrating factor for $(1)$.
Set of Condensation Points of Union is Union of Sets of Condensation Points
Let $T = \left({S, \tau}\right)$ be a topological space. Let $A, B$ be subsets of $S$. Then: :$\left({A \cup B}\right)^0 = A^0 \cup B^0$
Set of Condensation Points of Union is Union of Sets of Condensation Points/Lemma
Let $x$ be a point of $S$. Then: :if $x$ is condensation point of $A \cup B$, :then $x$ is condensation point of $A$ or $x$ is condensation point of $B$.
Integrating Factor for First Order ODE/Conclusion
Let the first order ordinary differential equation: :$(1): \quad \map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$ be non-homogeneous and not exact. Let $\map \mu {x, y}$be an integrating factor for $(1)$. If one of these is the case: :$\mu$ is a function of $x$ only :$\mu$ is a function of $y$ only :$\mu$ is a function of $x + y$ :$\mu$ is a function of $x y$ then: :$\mu = e^{\int \map f w \rd w}$ where $w$ depends on the nature of $\mu$.
First Order ODE/y dx + (x^2 y - x) dy = 0
The first order ODE: :$(1): \quad y \rd x + \paren {x^2 y - x} \rd y = 0$ has the general solution: :$\dfrac {y^2} 2 - \dfrac y x = C$
Set of Condensation Points of Countable Set is Empty
Let $T = \left({S, \tau}\right)$ be a topological space. Let $A$ be a subset of $S$. Then: :if $A$ is countable, :then $A^0 = \varnothing$.
Set of Condensation Points of Countable Set is Empty/Lemma
:if $A$ is countable, :then there exists no point $x$ of $S$ such that $x$ is a condensation point of $A$.
First Order ODE/(3 x^2 - y^2) dy - 2 x y dx = 0
The first order ODE: :$(1): \quad \paren {3 x^2 - y^2} \rd y - 2 x y \rd x = 0$ has the general solution: :$\dfrac 1 y - \dfrac {x^2} {y^3} = C$
First Order ODE/y dx + x dy + 3 x^3 y^4 dy
The first order ODE: :$(1): \quad y \rd x + x \rd y + 3 x^3 y^4 \rd y = 0$ has the general solution: :$-\dfrac 1 {2 x^2 y^2} + \dfrac {3 y^2} 2 = C$
Curved Mirror producing Parallel Rays is Paraboloid
Let $M$ be a curved mirror embedded in a real cartesian $3$- space. Let there be a source of light at the origin. Let $M$ reflect the light in a beam parallel to the $x$-axis. Then $M$ is the solid of revolution produced by rotating about the $x$-axis the parabola whose equation is: :$y^2 = 2 c x + c^2$
Quotient Rule for Differentials/Formulation 1
:$\map \d {\dfrac y x} = \dfrac {x \rd y - y \rd x} {x^2}$
Product Rule for Differentials
:$\map \d {x y} = x \rd y + y \rd x$
Differential of Sum of Squares
:$\map \d {x^2 + y^2} = 2 \paren {x \map \rd x + y \map \rd y}$
Differential of Arctangent of Quotient
:$\mathrm d \left({\arctan \dfrac x y}\right) = \dfrac{y \, \mathrm d x - x \, \mathrm d y} {x^2 + y^2}$
Differential of Logarithm of Quotient
:$\map \d {\ln \dfrac x y} = \dfrac {y \rd x - x \rd y} {x y}$
First Order ODE/y dx + (x^2 y - x) dy = 0/Proof 1
{{:First Order ODE/y dx + (x^2 y - x) dy = 0}} This can also be presented in the form: :$\dfrac {\d y} {\d x} + \dfrac y {x^2 y - x}$
First Order ODE/y dx - x dy = x y^3 dy
The first order ODE: :$(1): \quad y rd x - x \rd y = x y^3 \rd y$ has the general solution: :$\ln \dfrac x y = \dfrac {y^3} 3 + C$
First Order ODE/(x + y) dx = (x - y) dy/Proof 1
The first order ordinary differential equation: :$(1): \quad \paren {x + y} \rd x = \paren {x - y} \rd y$ is a homogeneous differential equation with solution: :$\arctan \dfrac y x = \ln \sqrt{x^2 + y^2} + C$
First Order ODE/x dy = (y + x^2 + 9 y^2) dx
The first order ODE: :$(1): \quad x \rd y = \paren {y + x^2 + 9 y^2} \rd x$ has the general solution: :$\map \arctan {\dfrac {3 y} x} = 3 x + C$
Linear First Order ODE/y' + (y over x) = 3 x
The linear first order ODE: :$\dfrac {\d y} {\d x} + \dfrac y x = 3 x$ has the general solution: :$x y = x^3 + C$ or: :$y = x^2 + \dfrac C x$
Linear First Order ODE/x y' - 3 y = x^4
The linear first order ODE: :$(1): \quad x \dfrac {\d y} {\d x} - 3y = x^4$ has the general solution: :$y = x^4 + \dfrac C {x^3}$
Linear First Order ODE/y' + y = 1 over (1 + exp 2 x)
The linear first order ODE: :$(1): \quad y' + y = \dfrac 1 {1 + e^{2 x} }$ has the general solution: :$y = e^{-x} \map \arctan {e^x} + C e^{-x}$
Linear First Order ODE/(1 + x^2) dy + 2 x y dx = cotangent x dx
The linear first order ODE: :$(1): \quad \paren {1 + x^2} \rd y + 2 x y \rd x = \cot x \rd x$ has the general solution: :$y = \dfrac {\map \ln {\sin x} } {1 + x^2} + \dfrac C {1 + x^2}$
Linear First Order ODE/(2 y - x^3) dx = x dy
The linear first order ODE: :$(1): \quad \paren {2 y - x^3} \rd x = x \rd y$ has the general solution: :$y = -x^3 + C x^2$
Solution to Bernoulli's Equation
'''Bernoulli's equation''': :$(1): \quad \dfrac {\d y} {\d x} + \map P x y = \map Q x y^n$ where: :$n \ne 0, n \ne 1$ has the general solution: :$\displaystyle \frac {\map \mu x} {y^{n - 1} } = \paren {1 - n} \int \map Q x \map \mu x \rd x + C$ where: :$\map \mu x = e^{\paren {1 - n} \int \map P x \rd x}$
Bernoulli's Equation/x y' + y = x^4 y^3
The first order ODE: :$(1): \quad x y' + y = x^4 y^3$ has the general solution: :$\dfrac 1 {y^2} = - x^4 + C x^2$
Topology Defined by Neighborhood System
Let $S$ be a set. Let $\family {\NN_x}_{x \mathop \in S}$ be an indexed family where $\NN_x$ is non-empty set of subsets of $S$. Assume that :$(\text N 1): \quad \forall x \in S, U \in \NN_x: x \in U$ :$(\text N 2): \quad \forall x \in S, U \in \NN_x, y \in U:\exists V \in \NN_y: V \subseteq U$ :$(\text N 3): \quad \forall x \in S, U_1, U_2 \in \NN_x: \exists U \in \NN_x: U \subseteq U_1 \cap U_2$ Then $T = \struct {S, \tau}$ is a topological space where: :$\tau = \displaystyle \set {\bigcup \GG: \GG \subseteq \bigcup_{x \mathop \in S} \NN_x}$ Moreover, $\family {\NN_x}_{x \mathop \in S}$ is a neighborhood system of $T$.
Orthogonal Trajectories/x + C exp -x
Consider the one-parameter family of curves: :$(1): \quad y = x + C e^{-x}$ Its family of orthogonal trajectories is given by the equation: :$x = y - 2 + C e^{-y}$ :600px
Solution of Second Order Differential Equation with Missing Dependent Variable
Let $\map f {x, y', y''} = 0$ be a second order ordinary differential equation in which the dependent variable $y$ is not explicitly present. Then $f$ can be reduced to a first order ordinary differential equation, whose solution can be determined.
Niemytzki Plane is Topology
Niemytzki plane is a topological space.
Solution of Second Order Differential Equation with Missing Independent Variable
Let $\map g {y, \dfrac {\d y} {\d x}, \dfrac {\d^2 y} {\d x^2} } = 0$ be a second order ordinary differential equation in which the independent variable $x$ is not explicitly present. Then $g$ can be reduced to a first order ordinary differential equation, whose solution can be determined.
Linear Second Order ODE/y'' + k^2 y = 0
The second order ODE: :$(1): \quad y'' + k^2 y = 0$ has the general solution: :$y = A \, \map \sin {k x + B}$ or can be expressed as: :$y = C_1 \sin k x + C_2 \cos k x$
Second Order ODE/y y'' + (y')^2 = 0
The second order ODE: :$(1): \quad y y'' + \paren {y'}^2 = 0$ has the general solution: :$y^2 = C_1 x + C_2$
Second Order ODE/x y'' = y' + (y')^3
The second order ODE: :$(1): \quad x y'' = y' + \paren {y'}^3$ has the general solution: :$x^2 + \paren {y - C_2}^2 = C_1^2$
Linear Second Order ODE/y'' - k^2 y = 0
The second order ODE: :$(1): \quad y'' - k^2 y = 0$ has the general solution: :$y = C_1 e^{k x} + C_2 e^{-k x}$
Second Order ODE/x^2 y'' = 2 x y' + (y')^2
The second order ODE: :$x^2 y'' = 2 x y' + \paren {y'}^2$ has the general solution: :$y = -\dfrac {x^2} 2 - C_1 x - {C_1}^2 \, \map \ln {x - C_1} + C_2$
Second Order ODE/y y'' = y^2 y' + (y')^2
The second order ODE: :$(1): \quad y y'' = y^2 y' + \paren {y'}^2$ subject to the initial conditions: :$y = -\dfrac 1 2$ and $y' = 1$ when $x = 0$ has the particular solution: :$2 y - 3 = 8 y \, \map \exp {\dfrac {3 x} 2}$
Second Order ODE/y'' = 1 + (y')^2
The second order ODE: :$(1): \quad y'' = 1 + \paren {y'}^2$ has the general solution: :$y = \map {\ln \sec} {x + C_1} + C_2$
Velocity of Bead on Brachistochrone
Consider a wire bent into the shape of an arc of a cycloid $C$ and inverted so that its cusps are uppermost and on the same horizontal line. Let $C$ be defined by Equation of Cycloid embedded in a cartesian plane: :$x = a \left({\theta - \sin \theta}\right)$ :$y = -a \left({1 - \cos \theta}\right)$ Let a bead $B$ be released from some point on the wire. Let $B$ slide without friction under the influence of a constant gravitational field exerting an acceleration $g$. Let $s_0$ be the arc length along the cycloid. Let $s$ be the arc length along the cycloid at any subsequent point in time. Then the speed $v$ of $B$ relative to $C$ is defined by the equation: :$4 a v^2 = g \left({ {s_0}^2 - s^2}\right)$
Open Ball is Subset of Open Ball
Let $M = \struct {A, d}$ be a metric space. Let $x, y$ be points of $A$. Then: : $\epsilon-\delta \ge \map d {x, y} \implies \map {B_\delta} y \subseteq \map {B_\epsilon} x$ where $\map {B_\epsilon} x$ denotes the open $\epsilon$-ball in $M = \struct {A, d}$.
Pursuit Curve of Boat in River
Consider a straight river $R$ whose parallel banks are aligned with the $y$-axis and the line $x = c$ of a cartesian plane. Let the current of $R$ have a constant and uniform speed $a$ in the negative $y$ direction. Let a boat $B$ be launched from the point $\left({c, 0}\right)$ and headed directly towards the origin with speed $b$ relative to the water. The path of $B$ is defined by the equation: :$c^k \left({y + \sqrt {x^2 + x^2} }\right) = x^{k + 1}$
Current in Electric Circuit/L, R, C in Series
Consider the electrical circuit $K$ consisting of: : a resistance $R$ : an inductance $L$ : a capacitance $C$ in series with a source of electromotive force $E$ which is a function of time $t$. :File:CircuitRLCseries.png The electric current $I$ in $K$ is given by the linear second order ODE: :$L \dfrac {\mathrm d^2 I} {\mathrm d t^2} + R \dfrac {\mathrm d I} {\mathrm d t} + \dfrac 1 C I = \dfrac {\mathrm d E} {\mathrm d t}$
Electric Charge in Electric Circuit/L, R, C in Series
Consider the electrical circuit $K$ consisting of: : a resistance $R$ : an inductance $L$ : a capacitance $C$ in series with a source of electromotive force $E$ which is a function of time $t$. :File:CircuitRLCseries.png The electric charge $Q$ in $K$ is given by the linear second order ODE: :$L \dfrac {\mathrm d^2 Q} {\mathrm d t^2} + R \dfrac {\mathrm d Q} {\mathrm d t} + \dfrac 1 C Q = E$
Current in Electric Circuit/L, R in Series
Consider the electrical circuit $K$ consisting of: :a resistance $R$ :an inductance $L$ in series with a source of electromotive force $E$ which is a function of time $t$. :File:CircuitRLseries.png The electric current $I$ in $K$ is given by the linear first order ODE: :$L \dfrac {\d I} {\d t} + R I = E$
Current in Electric Circuit/L, R in Series/Constant EMF at t = 0
Let the electric current flowing in $K$ at time $t = 0$ be $I_0$. Let a constant EMF $E_0$ be imposed upon $K$ at time $t = 0$. The electric current $I$ in $K$ is given by the equation: :$I = \dfrac {E_0} R + \paren {I_0 - \dfrac {E_0} R} e^{-R t / L}$