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# Separable

This section contains worked examples of the type of differential equation which can be solved by integration
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## 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 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:
and  at r=a and  at r=b

Workings






Substituting in the given values for V and r

and



Thus

Solution