Using the Method of Undetermined Coefficients
If you need to find particular solutions to nonhomogeneous differential equations, then you can start with the method of undetermined coefficients. Suppose you face the following nonhomogeneous differential equation:
The method of undetermined coefficients notes that when you find a candidate solution, y, and plug it into the left-hand side of the equation, you end up with g(x). Because g(x) is only a function of x, you can often guess the form of yp(x), up to arbitrary coefficients, and then solve for those coefficients by plugging yp(x) into the differential equation.
This method works because you’re dealing only with g(x), and the form of g(x) can often tell you what a particular solution looks like.