Laplace Transforms

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Laplace transforms are used to solve forced linear differential equations. In Section ?? we used the method of undetermined coefficients to solve forced equations when the forcing term $g(t)$ is of a special form, namely, when $g(t)$ is a linear combination of the functions $t^je^{\lambda t}$ , $t^je^{\sigma t}\sin (\tau t)$ , and $t^je^{\sigma t}\cos (\tau t)$ . Although Laplace transforms can be used to solve such systems as well, it is usually more efficient to use the method of undetermined coefficients when that method is applicable. The strength of the method of Laplace transforms is that it can be used to solve forced linear differential equations when the forcing term is more general. Specifically, Laplace transforms can be used to solve forced equations when the forcing term is either discontinuous (a step function) or an impulse function (a Dirac delta function).

The idea behind the Laplace transform method is that it is possible to transform any function so that there is a simple relationship between the transform of that function and the transform of its derivative. It is this observation, coupled with linearity, that leads to a useful and elegant method for solving linear, higher order, inhomogeneous differential equations. In Section ?? we discuss this property and the Laplace transform method for solving differential equations. In Section ?? we formally introduce the Laplace transform (as an improper integral), show that this definition satisfies the basic properties introduced in Section ??, and compute the Laplace transform for a variety of functions including step functions and Dirac delta functions.

When using the method of Laplace transforms to solve linear differential equations, it is immediately clear that this method demands the computation of partial fraction expansions. Some of the details of partial fraction expansions are discussed in Section ??.

In Section ?? we solve several linear differential equations with discontinuous forcing using the methods developed in the first three sections. RLC circuits provide an excellent example of a physical problem modeled by second order linear forced differential equations, and this model is discussed in Section ??. Discontinuous periodic forcing (AC current) also occurs in these models and their analysis is discussed in that section.