# Defining Homogeneous and Nonhomogeneous Differential Equations

In order to identify a nonhomogeneous differential equation, you first need to know what a homogeneous differential equation looks like. You also often need to solve one before you can solve the other.

**Homogeneous differential equations** involve only derivatives of *y* and terms involving *y*, and they’re set to 0, as in this equation:

**Nonhomogeneous differential equations** are the same as homogeneous differential equations, except they can have terms involving only *x* (and constants) on the right side, as in this equation:

You also can write nonhomogeneous differential equations in this format: *y*” + *p*(*x*)*y*‘ + *q*(*x*)*y* = *g*(*x*). The general solution of this nonhomogeneous differential equation is

In this solution, *c*_{1}*y*_{1}(*x*) + *c*_{2}*y*_{2}(*x*) is the general solution of the corresponding homogeneous differential equation:

And *y** _{p}*(

*x*) is a specific solution to the nonhomogeneous equation.