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Separable - Differential - Calculus - Maths Reference with Worked Examples

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Separable

This section contains worked examples of the type of differential equation which can be solved by integration

Separable Differential Equations

This section contains worked examples of the type of differential equation which can be solved by direct Integration.

Definition

Separable Differential Equations are differential equations which respect one of the following forms :
• $\inline&space;\displaystyle&space;\frac{dy}{dx}&space;=&space;F(x,y)$ where $\inline&space;F$ is a two variable function, also continuous.

• $\inline&space;\displaystyle&space;f(y)dy=g(x)dx$, where $\inline&space;f$ and $\inline&space;g$ are two real continuous functions.

Rational Functions

A rational function on $\inline&space;\mathhf{R}$ is a function $\inline&space;f:\mathhf{R}\to\mathhf{R}$ which can be expressed as $\inline&space;\displaystyle&space;f(x)=\frac{P(x)}{Q(x)}$ where $\inline&space;P,Q$ are two polynomials.
Example:
Example - Simple Differential Equation
Problem
Solve:
$\frac{d^2y}{dx^2}&space;=&space;2&space;+&space;\frac{1}{x}$

Workings
As the equation is of first order, integrate the function twice, i.e.
$\frac{dy}{dx}&space;=&space;2x&space;+&space;ln\,x&space;+&space;C$
and
$y&space;=&space;x^2&space;+&space;x\,ln\,x&space;+&space;Cx&space;+&space;K$
Solution
$y&space;=&space;x^2&space;+&space;x\,ln\,x&space;+&space;Cx&space;+&space;K$

Trigonometric Functions

A rational function on $\inline&space;\mathhf{R}$ is a function $\inline&space;f:\mathhf{R}\to\mathhf{R}$ which can be expressed as a combination of trigonometric functions ($\inline&space;sinx,cosx,tanx,cotanx$).
Example:
Example - Simple Cosine
Problem
$\frac{dy}{dx}=\frac{1}{cos^2&space;\tfrac{1}{2}x}$

Workings
This is the same as
$y=\int&space;\frac{1}{\cos^2&space;\tfrac{1}{2}x}dx$

which we integrate in the normal way to yield
$y=2\tan&space;\tfrac{1}{2}&space;x&space;+&space;C$
Solution
$y=2\tan&space;\tfrac{1}{2}&space;x&space;+&space;C$

Physics Examples

Example:
Example - Potential example
Problem
If a and b are the radii of concentric spherical conductors at potentials of $\inline&space;V_1$ respectively, then V is the potential at a distance r from the centre. Find the value of V if:
$\frac{d}{dr}\left(r^2&space;\frac{dV}{dr}&space;\right)=0}$
and $\inline&space;V=V_1$ at r=a and $\inline&space;V=0$ at r=b

Workings
$\frac{d}{dr}\left(r^2\frac{dV}{dr}&space;\right)=0$

$\therefore\;\;\;\;\;\;r^2\frac{dV}{dr}&space;\right)=A$

$\frac{dV}{A}=\frac{dr}{r^2}$

$\therefore\;\;\;\;\;\;\frac{V}{A}=-\frac{1}{r}\;+\;B$

Substituting in the given values for V and r
$\frac{V_1}{A}=-\frac{1}{a}+B$
and
$\frac{0}{A}=-\frac{1}{b}+B$
$\therefore\;\;\;\;\;\;B=\frac{1}{b}$
$\therefore\;\;\;\;\;\;\frac{V_1}{A}=-\frac{1}{a}+\frac{1}{b}=-\frac{b-a}{ab}$
Thus
$A=-\frac{abV_1}{b-a}$
Solution
$V=\left(\frac{1}{r}-\frac{1}{b}&space;\right)\left(\frac{abV_1}{b-a}&space;\right)$