• Overview

A linear differential equation is called homogeneous if the following condition is satisfied: If is a solution, so is , where is an arbitrary (non-zero) constant. Note that in order for this condition to hold, each term in a linear differential equation of the dependent variable y must contain y or any derivative of y.

A separable differential equation is any differential equation that we can write in the following form.

Note that in order for a differential equation to be separable all the y‘s in the differential equation must be multiplied by the derivative and all the x‘s in the differential equation must be on the other side of the equal sign. Solving separable differential equation is fairly easy. We first rewrite the differential equation as the following

Then you integrate both sides.

• Example

 

Example 2: Solve the equation

This equation is separable, since the variables can be separated:

The integral of the left‐hand side of this last equation is simply

and the integral of the right‐hand side is evaluated using integration by parts:

 

 

The solution of the differential equation is therefore

 

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