In this section we will see the first method which can be used to find an exact solution to a nonhomogeneous differential equation.

y′′ + p (t ) y′ + q (t ) y = g (t)

One of the major advantages of this method is that this reduces the problem down to an algebra problem. The algebra can find messy on occasion, but for mainly of the problems this will not be terribly difficult. The other nice thing about this method is that the complementary solution will not be explicitly needed, although as we will notice knowledge of the complementary solution will be required in some cases and so we'll generally determine that easily.

There are two drawbacks to this method. First, this will only work for a fairly small class of g(t)'s. The class of g(t)'s for that the method works, does comprise some of the more common functions, though, there are many functions out there for that undetermined coefficients simply won't work. Second, this is generally only helpful for constant coefficient differential equations.

The method is fairly simple. All that we require to do is look at g(t) and make a guess as to the form of Y_P(t) leaving the coefficients undetermined and thus the name of the method.  Plug the guess in the differential equation and notice if we can find out values of the coefficients. If we can find out values for the coefficients then we guessed properly, if we can't get values for the coefficients then we guessed imperfectly.

This is usually simple to notice this method in action quite than to try and describe it, therefore let's jump in some illustrations.