Let's here start thinking regarding that how to solve nonhomogeneous differential equations.  A second order, linear non-homogeneous differential equation is as

y′′ + p (t) y′ +q (t) y = g (t )     .....................(1)

Here g(t) is a non-zero function. Note that we didn't go along with constant coefficients here since everything that we're going to do under this section doesn't need it.  Also, we're using a coefficient of 1 on the second derivative just to create some of the work a little simple to write down. This is not needed to be a 1.

Before talking about how to resolve one of these we require to get some fundamentals out of the way that are the point of this section.

First, we will call

 y′′ + p (t ) y′ + q (t ) y = 0 (2)

It is the associated homogeneous differential equation to (1). Here, let's take a look at the subsequent theorem.