Separable Variables Differential Equations

 

Separable variables differential equations of first order have the following form:

As we can see, the independent variable x is totally separate from the unknown function y.

In order to solve this kind of differential equations, we integrate both sides of the equation:

We calculate the integration, then we solve the resulting algebraic equation and we get the unknown function y. If we know the initial conditions we can calculate C.

 

Example 1:

 where C and K are constant numbers.

If the initial condition is for x=0, then y=0:

 

Example 2:

A very common case of separable differential equations in Electrical and Electronic Engineering is when the rate of change of a variable is proportional to itself:

Where k is a constant number. This is solved as follows:

 Where:

is a constant number calculated from the initial conditions of the problem.