ABSTRACT
In this chapter, we examine how to solve linear equations in which there is a forcing function. In Sections 4.2 and 4.3, we consider two different methods for solving linear, constant-coefficient ODEs whose forcing functions are polynomials, exponentials, sinusoidal, or a combination of these. While the reader may read both of these sections, it is our intention that most will choose one of these methods for solving this type of equation (and thus skip the other section). In Section 4.4, we present an alternative method that is much quicker for forcing functions that are exponential or sinusoidal (viewed as complex exponentials) and thus might be ideal for engineering students; however, forcing functions that are multiplied by polynomials require a bit more work and one of the methods of Section 4.2 or 4.3 is needed to additionally solve an ODE that has polynomial-only forcing in order to complete the problem. Since most realistic engineering applications have forcing functions that involve decaying exponentials or sinusoidal functions, focusing mainly on the exponential or sinusoidal functions in Section 4.4 will likely be sufficient. These three sections are in contrast to Variation of Parameters, which does not require a prescribed type of forcing function nor does the equation need to have constant coefficients. Our reason for presenting all four methods is the greatly varied audience that is learning from this book, and each method has its place. We begin with a little background on nonhomogeneous equations.
