avewright/proofwiki-math huggingface dataset:
Theorem: Linear Second Order ODE/(x^2 + x) y'' + (2 - x^2) y' - (2 + x) y = 0

The second order ODE:
:$(1): \quad \paren {x^2 + x} y' ' + \paren {2 - x^2} y' - \paren {2 + x} y = 0$
has the general solution:
:$y = C_1 e^x + \dfrac {C_2} x$

Proof:
Note that:
:$\paren {x^2 + x} + \paren {2 - x^2} - \paren {2 + x} = 0$
so $\map {y_1} x$ such that $y_1 = {y_1}' = {y_1}' '$ satisfies $(1)$.
Hence:
:$y_1 = e^x$
is a particular solution of $(1)$.
$(1)$ can be expressed as:
:$(2): \quad y' ' + \dfrac {2 - x^2} {x^2 + x} y' - \dfrac {2 + x} {x^2 + x} y = 0$
which is in the form:
:$y' ' + \map P x y' + \map Q x y = 0$
where:
:$\map P x = \dfrac {2 - x^2} {x^2 + x}$
:$\map Q x = \dfrac {2 + x} {x^2 + x}$
From Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another:
:$\map {y_2} x = \map v x \, \map {y_1} x$
where:
:$\ds v = \int \dfrac 1 { {y_1}^2} e^{-\int P \rd x} \rd x$
is also a particular solution of $(1)$.
We have that:
{{begin-eqn}}
{{eqn | l = \int P \rd x
      | r = \int \paren {\dfrac {2 - x^2} {x^2 + x} } \rd x
      | c = 
}}
{{eqn | r = \int \dfrac 2 {x \paren {x + 1} } \rd x - \int \paren {\dfrac x {x + 1} } \rd x
      | c = 
}}
{{eqn | r = 2 \, \map \ln {\frac x {x + 1} } - \int \paren {\dfrac x {x + 1} } \rd x
      | c = Primitive of $\dfrac 1 {x \paren {a x + b} }$
}}
{{eqn | r = 2 \, \map \ln {\frac x {x + 1} } - x + \map \ln {x + 1}
      | c = Primitive of $\dfrac x {a x + b}$
}}
{{eqn | r = \map \ln {\frac {x^2} {x + 1} } - x
      | c = 
}}
{{eqn | r = \map \ln {\frac {e^{-x} x^2} {x + 1} }
      | c = 
}}
{{eqn | r = -\map \ln {\frac {e^x \paren {x + 1} } {x^2} }
      | c = 
}}
{{eqn | ll= \leadsto
      | l = e^{-\int P \rd x}
      | r = e^{\map \ln {\frac {e^x \paren {x + 1} } {x^2} } }
      | c = 
}}
{{eqn | r = \frac {e^x \paren {x + 1} } {x^2}
      | c =  
}}
{{end-eqn}}
Hence:
{{begin-eqn}}
{{eqn | l = v
      | r = \int \dfrac 1 { {y_1}^2} e^{-\int P \rd x} \rd x
      | c = Definition of $v$
}}
{{eqn | r = \int e^{-2 x} \paren {\frac {e^x \paren {x + 1} } {x^2} } \rd x
      | c = 
}}
{{eqn | r = \int \frac {e^{-x} \paren {x + 1} } {x^2} \rd x
      | c = 
}}
{{eqn | r = \int \frac {e^{-x} } x \rd x + \int \frac {e^{-x} } {x^2} \rd x
      | c = 
}}
{{eqn | r = \int \frac {e^{-x} } x \rd x + \frac {-e^{-x} } x - \int \frac {e^{-x} } x \rd x
      | c = Primitive of $\dfrac {e^{a x} } {x^n}$
}}
{{eqn | r = -\frac {e^{-x} } x
      | c = 
}}
{{end-eqn}}
and so:
{{begin-eqn}}
{{eqn | l = y_2
      | r = v y_1
      | c = Definition of $y_2$
}}
{{eqn | r = \paren {-\frac {e^{-x} } x} e^x
      | c = 
}}
{{eqn | r = -\frac 1 x
      | c = 
}}
{{end-eqn}}
From Two Linearly Independent Solutions of Homogeneous Linear Second Order ODE generate General Solution:
:$y = C_1 e^x + \dfrac {C_2} x$
{{qed}}
Theorem: Directed in Join Semilattice

Let $\struct {S, \preceq}$ be a join semilattice.
Let $H$ be a non-empty lower section of $S$.
Then $H$ is directed {{iff}}
:$\forall x, y \in H: x \vee y \in H$

Proof:
=== Sufficient Condition ===
Let us assume that
:$H$ is directed.
Let $x, y \in H$.
By definition of directed:
:$\exists z \in H: x \preceq z \land y \preceq z$
By definition
:$z$ is upper bound of $\set {x, y}$
By definitions of supremum and join:
:$x \vee y = \sup \set {x, y} \preceq z$
Thus by definition of lower section:
:$x \vee y \in H$
{{qed|lemma}}
Theorem: Period of Oscillation of Underdamped System is Regular

Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form:
:$\dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$
for $a, b \in \R_{>0}$.
Let $b < a$, so as to make $S$ underdamped.
Then the period of its movement is well-defined, in the sense that its zeroes are regularly spaced, and given by:
:$T = \dfrac {2 \pi} {\sqrt {a^2 - b^2} }$

Proof:
Let the position of $S$ be described in the canonical form:
:$(1): \quad x = \dfrac {x_0 \, a} \alpha e^{-b t} \map \cos {\alpha t - \theta}$
where $\alpha = \sqrt {a^2 - b^2}$.
The zeroes of $(1)$ occur exactly where:
:$\map \cos {\alpha t - \theta} = 0$
Thus the period $T$ of $\map \cos {\alpha t - \theta}$ is given by:
:$\alpha T = 2 \pi$
and so:
:$T = \dfrac {2 \pi} {\sqrt {a^2 - b^2} }$
{{qed}}
Theorem: Natural Frequency of Underdamped System

Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form:
:$\dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$
for $a, b \in \R_{>0}$.
Let $b < a$, so as to make $S$ underdamped.
Then the natural frequency of $S$ is given by:
:$\nu = \dfrac {\sqrt {a^2 - b^2} } {2 \pi}$

Proof:
Let the position of $S$ be described in the canonical form:
:$(1): \quad x = \dfrac {x_0 \, a} \alpha e^{-b t} \map \cos {\alpha t - \theta}$
where $\alpha = \sqrt {a^2 - b^2}$.
From Period of Oscillation of Underdamped System is Regular:
:$T = \dfrac {2 \pi} {\sqrt {a^2 - b^2} }$
where $T$ is the period of the movement of $S$.
By definition of natural frequency:
:$\nu = \dfrac 1 T = \dfrac {\sqrt {a^2 - b^2} } {2 \pi}$
{{qed}}
Theorem: Linear Second Order ODE/y'' + 2 b y' + a^2 y = 0/b less than a

The second order ODE:
:$(1): \quad y' ' + 2 b y' + a^2 y = 0$ where $b^2 < a^2$
has the general solution:
:$y = e^{-b x} \paren {C_1 \, \map \cos {\sqrt {a^2 - b^2} } x + C_2 \, \map \sin {\sqrt {a^2 - b^2} } x}$

Proof:
It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad: m^2 + 2 b m + a^2 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = - b + \sqrt {b^2 - a^2}$
:$m_2 = - b - \sqrt {b^2 - a^2}$
As $b^2 < a^2$, this converts into:
:$m_1 = - b + i \sqrt {a^2 - b^2}$
:$m_2 = - b - i \sqrt {a^2 - b^2}$
These are complex and unequal.
So from Solution of Constant Coefficient Homogeneous LSOODE, the general solution of $(1)$ is:
:$y = e^{-b x} \paren {C_1 \, \map \cos {\sqrt {a^2 - b^2} } x + C_2 \, \map \sin {\sqrt {a^2 - b^2} } x}$
{{qed}}
Category:Examples of Constant Coefficient Homogeneous LSOODEs
i9bedg8zmjiri836fw17pz0fibadew8
Theorem: Condition for Resonance in Forced Vibration of Underdamped System

Consider a physical system $S$ whose behaviour is defined by the second order ODE:
:$(1): \quad \dfrac {\d^2 y} {\d x^2} + 2 b \dfrac {\d y} {\d x} + a^2 x = K \cos \omega x$
where:
:$K \in \R: k > 0$
:$a, b \in \R_{>0}: b < a$
Then $S$ is in resonance when:
:$\omega = \sqrt {a^2 - 2 b^2}$
and thus the resonance frequency is:
:$\nu_R = \dfrac {\sqrt {a^2 - 2 b^2} } {2 \pi}$
This resonance frequency exists only when $a^2 - 2 b^2 > 0$.

Proof:
From Linear Second Order ODE: $y' ' + 2 b y' + a^2 y = K \cos \omega x$ where $b < a$ the general solution of $(1)$ is:
:$(2): \quad y = e^{-b x} \paren {C_1 \cos \alpha x + C_2 \sin \alpha x} + \dfrac K {\sqrt {4 b^2 \omega^2 + \paren {a^2 - \omega^2}^2} } \map \cos {\omega x - \phi}$
where:
:$\alpha = \sqrt {a^2 - b^2}$
:$\phi = \map \arctan {\dfrac {2 b \omega} {a^2 - \omega^2} }$
By definition, $S$ is '''in resonance''' when the amplitude of the steady-state component of $(1)$ is at a maximum.
We are to determine the value of $\omega$ when this is the case.
Consider the steady-state component of $(2)$:
:$y_s = \dfrac K {\sqrt {4 b^2 \omega^2 + \paren {a^2 - \omega^2}^2} } \map \cos {\omega x - \phi}$
By definition, the amplitude of $y_s$ is:
:$A = \dfrac K {\sqrt {4 b^2 \omega^2 + \paren {a^2 - \omega^2}^2} }$
We have that both $a$ and $b$ are greater than $0$.
It is also assumed that $\omega$ is also greater than $0$.
Thus:
:$4 b^2 \omega^2 > 0$
:$a^2 > 0$
:$\omega^2 > 0$
Then $y_s$ is at a maximum when:
:$\map f \omega = 4 b^2 \omega^2 + \paren {a^2 - \omega^2}^2$
is at a minimum.
Differentiating $\map f \omega$ {{WRT|Differentiation}} $\omega$:
{{begin-eqn}}
{{eqn | l = \map {f'} \omega
      | r = 8 b^2 \omega + 2 \paren {a^2 - \omega^2} \paren {-2 \omega}
      | c = 
}}
{{eqn | r = 8 b^2 \omega - 4 a^2 \omega + 4 \omega^3
      | c = 
}}
{{eqn | r = 4 \omega \paren {2 b^2 - a^2 + \omega^2}
      | c = 
}}
{{end-eqn}}
Setting $\map {f'} \omega = 0$:
{{begin-eqn}}
{{eqn | l = 4 \omega \paren {2 b^2 - a^2 + \omega^2}
      | r = 0
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \omega
      | r = 0
      | c = 
}}
{{eqn | lo= \lor
      | l = \omega^2
      | r = a^2 - 2 b^2
      | c = 
}}
{{end-eqn}}
As $\omega > 0$ the first of these cannot apply.
If $a^2 < 2 b^2$ then $a^2 - 2 b^2 < 0$ and so $\omega^2$ has no real roots.
Hence the resonance frequency is given by:
:$\omega = \sqrt {a^2 - 2 b^2}$
{{iff}} $a^2 > 2 b^2$.
{{qed}}
Theorem: Filtered in Meet Semilattice

Let $\struct {S, \preceq}$ be a meet semilattice.
Let $H$ be a non-empty upper section of $S$.
Then $H$ is filtered {{iff}}
:$\forall x, y \in H: x \wedge y \in H$

Proof:
This follows by {{mutatis}} of the proof of Directed in Join Semilattice.
{{qed}}
Theorem: Directed in Join Semilattice with Finite Suprema

Let $\struct {S, \preceq}$ be a join semilattice.
Let $H$ be a non-empty lower section of $S$.
Then $H$ is directed {{iff}}
:for every non-empty finite subset $A$ of $H$, $\sup A \in H$

Proof:
=== Sufficient Condition ===
Let us assume that
:$H$ is directed.
Let $A$ be a non-empty finite subset of $H$.
By Directed iff Finite Subsets have Upper Bounds:
:$\exists h \in H: \forall a \in A: a \preceq h$
By definition
:$z$ is upper bound of $A$
By Existence of Non-Empty Finite Suprema in Join Semilattice:
:$\sup A$ exists in $\struct {S, \preceq}$
By definition of supremum:
:$\sup A \preceq h$
Thus by definition of lower section:
:$\sup A \in H$
{{qed|lemma}}
Theorem: Resonance Frequency is less than Natural Frequency

Consider a physical system $S$ whose behaviour is defined by the second order ODE:
:$(1): \quad \dfrac {\d^2 y} {\d x^2} + 2 b \dfrac {\d y} {\d x} + a^2 x = K \cos \omega x$
where:
:$K \in \R: k > 0$
:$a, b \in \R_{>0}: b < a$
Then the resonance frequency of $S$ is smaller than the natural frequency of the associated second order ODE:
:$(2): \quad \dfrac {\d^2 y} {\d x^2} + 2 b \dfrac {\d y} {\d x} + a^2 x = 0$

Proof:
From Natural Frequency of Underdamped System, the natural frequency of $(2)$ is:
:$\nu = \dfrac {\sqrt {a^2 - b^2} } {2 \pi}$
From Condition for Resonance in Forced Vibration of Underdamped System, the resonance frequency of $S$ is:
:$\omega = \sqrt {a^2 - 2 b^2}$
:$\nu_R = \dfrac {\sqrt {a^2 - 2 b^2} } {2 \pi}$
We have that:
{{begin-eqn}}
{{eqn | l = \dfrac \nu {\nu_R}
      | r = \dfrac {\sqrt {a^2 - b^2} } {\sqrt {a^2 - 2 b^2} }
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \dfrac {\nu^2} { {\nu_R}^2}
      | r = \dfrac {a^2 - b^2} {a^2 - 2 b^2}
      | c = 
}}
{{eqn | r = \dfrac {a^2 - 2 b^2 + b^2} {a^2 - 2 b^2}
      | c = 
}}
{{eqn | r = 1 + \dfrac {b^2} {a^2 - 2 b^2}
      | c = 
}}
{{end-eqn}}
Thus $\nu_R < \nu$ {{iff}} $a^2 - 2 b^2 > 0$.
But from Condition for Resonance in Forced Vibration of Underdamped System, in order for there to be resonance, $a^2 - 2 b^2 > 0$.
The two conditions are compatible, and so hence the result.
{{qed}}
Theorem: Filtered in Meet Semilattice with Finite Infima

Let $\struct {S, \preceq}$ be a meet semilattice.
Let $H$ be a non-empty upper section of $S$.
Then $H$ is filtered {{iff}}
:for every non-empty finite subset $A$ of $H$, $\inf A \in H$

Proof:
This follows by {{mutatis}} of the proof of Directed in Join Semilattice with Finite Suprema.
{{qed}}
Theorem: Ratio of Successive Local Maxima for Underdamped Free Vibration

Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form:
:$(1): \quad \dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$
for $a, b \in \R_{>0}$.
Let $b < a$, so as to make $S$ underdamped.
:600px
Let $T$ be the period of oscillation of $S$.
Let $x_1$ and $x_2$ be successive local maxima of $x$.
Then:
:$\dfrac {x_1} {x_2} = e^{b T}$

Proof:
Let the position of $S$ be described in the canonical form:
:$(1): \quad x = \dfrac {x_0 \, a} \alpha e^{-b t} \map \cos {\alpha t - \theta}$
where:
:$\alpha = \sqrt {a^2 - b^2}$
:$\theta = \map \arctan {\dfrac b \alpha}$
From Period of Oscillation of Underdamped System is Regular, the period of oscillation $T$ is given by:
:$T = \dfrac {2 \pi} {a^2 - b^2}$
From Interval between Local Maxima for Underdamped Free Vibration, the local maxima of $x$ occur at:
:$t = 0, T, 2 T, \ldots$
Let $x_1$ occur at $n T$.
Thus $x_2$ occurs at $\paren {n + 1} T$.
Differentiating {{WRT|Differentiation}} $t$:
{{begin-eqn}}
{{eqn | l = x_1
      | r = \dfrac {x_0 \, a} \alpha e^{-b n T} \map \cos {\alpha n T - \theta}
      | c = 
}}
{{eqn | l = x_2
      | r = \dfrac {x_0 \, a} \alpha e^{-b \paren {n + 1} T} \map \cos {\alpha \paren {n + 1} T - \theta}
      | c = 
}}
{{end-eqn}}
As $T = \dfrac {2 \pi} {a^2 - b^2} = \dfrac {2 \pi} \alpha$, we have:
{{begin-eqn}}
{{eqn | l = \map \cos {\alpha \paren {n + 1} T - \theta}
      | r = \map \cos {\alpha \paren {n + 1} \dfrac {2 \pi} \alpha - \theta}
      | c = 
}}
{{eqn | r = \map \cos {\paren {n + 1} 2 \pi - \theta}
      | c = 
}}
{{eqn | r = \map \cos {n 2 \pi - \theta}
      | c = 
}}
{{eqn | r = \map \cos {\alpha n \dfrac {2 \pi} \alpha  - \theta}
      | c = 
}}
{{eqn | r = \map \cos {\alpha n T  - \theta}
      | c = 
}}
{{end-eqn}}
Hence:
{{begin-eqn}}
{{eqn | l = \dfrac {x_1} {x_2}
      | r = \dfrac {\dfrac {x_0 \, a} \alpha e^{-b n T} \map \cos {\alpha n T - \theta} } {\dfrac {x_0 \, a} \alpha e^{-b \paren {n + 1} T} \map \cos {\alpha \paren {n + 1} T - \theta} }
      | c = 
}}
{{eqn | r = \dfrac {e^{-b n T} \map \cos {\alpha n T - \theta} } {e^{-b \paren {n + 1} T} \map \cos {\alpha n T - \theta} }
      | c = 
}}
{{eqn | r = \dfrac {e^{-b n T} } {e^{-b \paren {n + 1} T} }
      | c = 
}}
{{eqn | r = e^{-b n T + b \paren {n + 1} T}
      | c = 
}}
{{eqn | r = e^{b T}
      | c = 
}}
{{end-eqn}}
as required.
{{qed}}
Theorem: Motion of Particle in Polar Coordinates

Consider a particle $p$ of mass $m$ moving in the plane under the influence of a force $\mathbf F$.
Let the position of $p$ at time $t$ be given in polar coordinates as $\polar {r, \theta}$.
Let $\mathbf F$ be expressed as:
:$\mathbf F = F_r \mathbf u_r + F_\theta \mathbf u_\theta$
where:
:$\mathbf u_r$ is the unit vector in the direction of the radial coordinate of $p$
:$\mathbf u_\theta$ is the unit vector in the direction of the angular coordinate of $p$
:$F_r$ and $F_\theta$ are the magnitudes of the components of $\mathbf F$ in the directions of $\mathbf u_r$ and $\mathbf u_\theta$ respectively.
Then the second order ordinary differential equations governing the motion of $m$ under the force $\mathbf F$ are:
{{begin-eqn}}
{{eqn | l = F_\theta
      | r = m \paren {r \dfrac {\d^2 \theta} {\d t^2} + 2 \dfrac {\d r} {\d t} \dfrac {\d \theta} {\d t} }
      | c =
}}
{{eqn | l = F_r
      | r = m \paren {\dfrac {\d^2 r} {\d t^2} - r \paren {\dfrac {\d \theta} {\d t} }^2}
      | c =
}}
{{end-eqn}}

Proof:
Let the radius vector $\mathbf r$ from the origin to $p$ be expressed as:
:$(1): \quad \mathbf r = r \mathbf u_r$
From Velocity Vector in Polar Coordinates, the velocity $\mathbf v$ of $p$ can be expressed in vector form as:
:$\mathbf v = r \dfrac {\d \theta} {\d t} \mathbf u_\theta + \dfrac {\d r} {\d t} \mathbf u_r$
From Acceleration Vector in Polar Coordinates, the the acceleration $\mathbf a$ of $p$ can be expressed as:
:$\mathbf a = \paren {r \dfrac {\d^2 \theta} {\d t^2} + 2 \dfrac {\d r} {\d t} \dfrac {\d \theta} {\d t} } \mathbf u_\theta + \paren {\dfrac {\d^2 r} {\d t^2} - r \paren {\dfrac {\d \theta} {\d t} }^2} \mathbf u_r$
We have:
:$\mathbf F = F_r \mathbf u_r + F_\theta \mathbf u_\theta$
and from Newton's Second Law of Motion:
:$\mathbf F = m \mathbf a$
Hence:
:$\mathbf F = m \paren {r \dfrac {\d^2 \theta} {\d t^2} + 2 \dfrac {\d r} {\d t} \dfrac {\d \theta} {\d t} } \mathbf u_\theta + m \paren {\dfrac {\d^2 r} {\d t^2} - r \paren {\dfrac {\d \theta} {\d t} }^2} \mathbf u_r$
Equating components:
{{begin-eqn}}
{{eqn | l = F_r \mathbf u_r
      | r = m \paren {\dfrac {\d^2 r} {\d t^2} - r \paren {\dfrac {\d \theta} {\d t} }^2} \mathbf u_r
      | c =
}}
{{eqn | l = F_\theta \mathbf u_\theta
      | r = m \paren {r \dfrac {\d^2 \theta} {\d t^2} + 2 \dfrac {\d r} {\d t} \dfrac {\d \theta} {\d t} } \mathbf u_\theta
      | c =
}}
{{end-eqn}}
Hence the result:
{{begin-eqn}}
{{eqn | l = F_\theta
      | r = m \paren {r \dfrac {\d^2 \theta} {\d t^2} + 2 \dfrac {\d r} {\d t} \dfrac {\d \theta} {\d t} }
      | c =
}}
{{eqn | l = F_r
      | r = m \paren {\dfrac {\d^2 r} {\d t^2} - r \paren {\dfrac {\d \theta} {\d t} }^2}
      | c =
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Angular Component under Central Force

Let a point mass $p$ of mass $m$ be under the influence of a central force $\mathbf F$.
Let the position of $p$ at time $t$ be given in polar coordinates as $\polar {r, \theta}$.
Let $\mathbf r$ be the radius vector from the origin to $p$.
Then the rate of change of the angular coordinate of $p$ is inversely proportional to the square of the radial coordinate of $p$.

Proof:
Let $\mathbf F$ be expressed as:
:$\mathbf F = F_r \mathbf u_r + F_\theta \mathbf u_\theta$
where:
:$\mathbf u_r$ is the unit vector in the direction of the radial coordinate of $p$
:$\mathbf u_\theta$ is the unit vector in the direction of the angular coordinate of $p$
:$F_r$ and $F_\theta$ are the magnitudes of the components of $\mathbf F$ in the directions of $\mathbf u_r$ and $\mathbf u_\theta$ respectively.
From Motion of Particle in Polar Coordinates, the second order ordinary differential equations governing the motion of $m$ under the force $\mathbf F$ are:
{{begin-eqn}}
{{eqn | l = F_\theta
      | r = m \paren {r \dfrac {\d^2 \theta} {\d t^2} + 2 \dfrac {\d r} {\d t} \dfrac {\d \theta} {\d t} }
      | c = 
}}
{{eqn | l = F_r
      | r = m \paren {\dfrac {\d^2 r} {\d t^2} - r \paren {\dfrac {\d \theta} {\d t} }^2}
      | c = 
}}
{{end-eqn}}
However, we are given that $\mathbf F$ is a central force.
:600px
Thus:
{{begin-eqn}}
{{eqn | l = F_\theta
      | r = 0
      | c = 
}}
{{eqn | ll= \leadsto
      | l = r \dfrac {\d^2 \theta} {\d t^2} + 2 \dfrac {\d r} {\d t} \dfrac {\d \theta} {\d t}
      | r = 0
      | c = 
}}
{{eqn | ll= \leadsto
      | l = r^2 \dfrac {\d^2 \theta} {\d t^2} + 2 r \dfrac {\d r} {\d t} \dfrac {\d \theta} {\d t}
      | r = 0
      | c = multiplying through by $r$
}}
{{eqn | ll= \leadsto
      | l = \dfrac \d {\d t} \paren {r^2 \dfrac {\d \theta} {\d t} }
      | r = 0
      | c = Product Rule for Derivatives
}}
{{eqn | ll= \leadsto
      | l = r^2 \dfrac {\d \theta} {\d t}
      | r = h
      | c = Derivative of Constant
}}
{{end-eqn}}
for some constant $h$.
That is:
:$\dfrac {\d \theta} {\d t} = \dfrac h {r^2}$
Hence the result, by definition of inverse proportion.
{{qed}}
Theorem: Linear Second Order ODE/y'' + y = K

The second order ODE:
:$(1): \quad y' ' + y = K$
has the general solution:
:$y = C_1 \sin x + C_2 \cos x + K$

Proof:
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:
:$p = 0$
:$q = 1$
:$\map R x = K$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$(2): \quad y' ' + y = 0$
From Linear Second Order ODE: $y' ' + y = 0$, this has the general solution:
:$y_g = C_1 \sin x + C_2 \cos x$
We have that:
:$\map R x = K$
So from the Method of Undetermined Coefficients for Polynomial:
:$y_p = A K$
where $A$ is to be determined.
Hence:
{{begin-eqn}}
{{eqn | l = y_p
      | r = A K
      | c = 
}}
{{eqn | ll= \leadsto
      | l = {y_p}'
      | r = 0
      | c = Derivative of Constant
}}
{{eqn | ll= \leadsto
      | l = {y_p}' '
      | r = 0
      | c = Derivative of Constant
}}
{{end-eqn}}
Substituting into $(1)$:
{{begin-eqn}}
{{eqn | l = A K
      | r = K
      | c = 
}}
{{eqn | ll= \leadsto
      | l = A
      | r = 1
}}
{{end-eqn}}
Our work is done.
So from General Solution of Linear 2nd Order ODE from Homogeneous 2nd Order ODE and Particular Solution:
:$y = y_g + y_p = C_1 \sin x + C_2 \cos x + K$
{{qed}}
Category:Examples of Constant Coefficient LSOODEs
Category:Examples of Method of Undetermined Coefficients
j3iz701ltzacxswi7u6249dpz99xyuz
Theorem: Velocity Vector in Polar Coordinates

Consider a particle $p$ moving in the plane.
Let the position of $p$ at time $t$ be given in polar coordinates as $\left\langle{r, \theta}\right\rangle$.
Then the velocity $\mathbf v$ of $p$ can be expressed as:
:$\mathbf v = r \dfrac {\d \theta} {\d t} \mathbf u_\theta + \dfrac {\d r} {\d t} \mathbf u_r$
where:
:$\mathbf u_r$ is the unit vector in the direction of the radial coordinate of $p$
:$\mathbf u_\theta$ is the unit vector in the direction of the angular coordinate of $p$

Proof:
Let the radius vector $\mathbf r$ from the origin to $p$ be expressed as:
:$(1): \quad \mathbf r = r \mathbf u_r$
:600px
From Derivatives of Unit Vectors in Polar Coordinates:
{{begin-eqn}}
{{eqn | n = 2
      | l = \dfrac {\d \mathbf u_r} {\d \theta}
      | r = \mathbf u_\theta
      | c = 
}}
{{eqn | n = 3
      | l = \dfrac {\d \mathbf u_\theta} {\d \theta}
      | r = -\mathbf u_r
      | c = 
}}
{{end-eqn}}
The velocity of $p$ is by definition the rate of change in its position:
{{begin-eqn}}
{{eqn | l = \mathbf v
      | r = \dfrac {\d \mathbf r} {\d t}
      | c = 
}}
{{eqn | r = r \dfrac {\d \mathbf u_r} {\d t} + \mathbf u_r \dfrac {\d r} {\d t}
      | c = from $(1)$ and Product Rule for Derivatives
}}
{{eqn | r = r \dfrac {\d \mathbf u_r} {\d \theta} \dfrac {\d \theta} {\d t} + \mathbf u_r \dfrac {\d r} {\d t}
      | c = Chain Rule for Derivatives
}}
{{eqn | r = r \dfrac {\d \theta} {\d t} \mathbf u_\theta + \dfrac {\d r} {\d t} \mathbf u_r
      | c = substituting from $(2)$ and $(3)$
}}
{{end-eqn}}
{{qed}}
Theorem: Derivatives of Unit Vectors in Polar Coordinates

Consider a particle $p$ moving in the plane.
Let the position of $p$ be given in polar coordinates as $\polar {r, \theta}$.
Let:
:$\mathbf u_r$ be the unit vector in the direction of the radial coordinate of $p$
:$\mathbf u_\theta$ be the unit vector in the direction of the angular coordinate of $p$
Then the derivative of $\mathbf u_r$ and $\mathbf u_\theta$ {{WRT}} $\theta$ can be expressed as:
{{begin-eqn}}
{{eqn | l = \dfrac {\d \mathbf u_r} {\d \theta}
      | r = \mathbf u_\theta
}}
{{eqn | l = \dfrac {\d \mathbf u_\theta} {\d \theta}
      | r = -\mathbf u_r
}}
{{end-eqn}}

Proof:
By definition of sine and cosine:
{{begin-eqn}}
{{eqn | n = 1
      | l = \mathbf u_r
      | r = \mathbf i \cos \theta + \mathbf j \sin \theta
}}
{{eqn | n = 2
      | l = \mathbf u_\theta
      | r = -\mathbf i \sin \theta + \mathbf j \cos \theta
}}
{{end-eqn}}
where $\mathbf i$ and $\mathbf j$ are the unit vectors in the $x$-axis and $y$-axis respectively.
:600px
Differentiating $(1)$ and $(2)$ {{WRT|Differentiation}} $\theta$ gives:
{{begin-eqn}}
{{eqn | l = \dfrac {\d \mathbf u_r} {\d \theta}
      | r = -\mathbf i \sin \theta + \mathbf j \cos \theta
      | c = 
}}
{{eqn | r = \mathbf u_\theta
      | c = 
}}
{{eqn | l = \dfrac {\d \mathbf u_\theta} {\d \theta}
      | r = -\mathbf i \cos \theta - \mathbf j \sin \theta
      | c = 
}}
{{eqn | r = -\mathbf u_r
      | c = 
}}
{{end-eqn}}
{{qed}}