# Ordinary Differential Equations/Nonhomogeneous second order equations:Method of undetermined coefficients

Consider a differencial equation of the form

Clerarly, this is not homogeneous, as .

So, to solve this, we first proceeed as normal, but assume that the equation is homogeneous; set for now. Then the first part of the solution pans like

Now we need to find the *particular* integral. To do this, make an appropriate substitution that relates to what is. For instance, if , then take substitution .
As and are multiples of in this case, you'll simply get a linear equation in . Then just plug the value of in the equation.

Hence the solution is

**y = general solution + particular integral**.

There is one important caveat which you should be aware though. In the previous example for instance, if the general solution already had , the substitution *cannot* be , as the particular integral cannot be equal to the general solution. In such cases, you need to take the substitution as .

## ExampleEdit

Solve the differential equation

Given that

### SolutionEdit

Take . Then

Hence the general form of the equation becomes

Then

. Then

Hence the final equation is