We explore the solution of nonhomogeneous linear equations with other forcing functions.

The Method of Undetermined Coefficients II

In this section we consider the constant coefficient equation

where and are real numbers, , and and are polynomials. We want to find a particular solution of (eq:5.5.1). As in the previous module, the procedure that we will use is called the method of undetermined coefficients.

Forcing Functions Without Exponential Factors

We begin with the case where in (eq:5.5.1); thus, we we want to find a particular solution of

where and are polynomials.

Differentiating and yields and This implies that if where and are polynomials, then where and are polynomials with coefficients that can be expressed in terms of the coefficients of and . This suggests that we try to choose and so that and , respectively. Then will be a particular solution of (eq:5.5.2). The next theorem tells us how to choose the proper form for .

A Useful Observation

In (eq:5.5.9), (eq:5.5.10), and (eq:5.5.11) the polynomials multiplying can be obtained by replacing , and by , , , and , respectively, in the polynomials multiplying .

We won’t use this theorem in our examples, but we recommend that you use it to check your manipulations when you work on the exercises.

Forcing Functions with Exponential Factors

To find a particular solution of

when , we recall from the previous module that substituting into (eq:5.5.15) will produce a constant coefficient equation for with the forcing function . We can find a particular solution of this equation by the procedure that we used in Examples example:5.5.1example:5.5.4. Then is a particular solution of (eq:5.5.15).

You can also find a particular solution of (eq:5.5.20) by substituting for in (eq:5.5.20) and equating the coefficients of , , , and in the resulting expression for with the corresponding coefficients on the right side of (eq:5.5.20). This leads to the same system (eq:5.5.22) of equations for , , , and that we obtained in Example example:5.5.6. However, if you try this approach you’ll see that deriving (eq:5.5.22) this way is much more tedious than the way we did it in Example example:5.5.6.

Text Source

Trench, William F., ”Elementary Differential Equations” (2013). Faculty Authored and Edited Books & CDs. 8. (CC-BY-NC-SA)