# 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 :- where is a two variable function, also continuous.

- , where and are two real continuous functions.

### Rational Functions

- A
**rational**function on is a function which can be expressed as where are two polynomials.

Example:

##### Example - Simple Differential Equation

Problem

Solve:

Workings

As the equation is of first order, integrate the function twice, i.e.
and

Solution

### Trigonometric Functions

- A
**rational**function on is a function which can be expressed as a combination of trigonometric functions ().

Example:

##### Example - Simple Cosine

Problem

Workings

This is the same as
which we integrate in the normal way to yield

Solution

### Physics Examples

Example:

##### Example - Potential example

Problem

If
and at r=a and at r=b

*a*and*b*are the radii of concentric spherical conductors at potentials of 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 \frac{dV}{dr} \right)=0}

Workings

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

*V*and

*r*and Thus

Solution