The previous three chapters were designed to help you understand the meaning and the method of the Laplace transform and its relation to the Fourier transform (), to show the Laplace transform of a few basic functions (), and to demonstrate some of the properties that make the Laplace transform useful (). In this chapter, you will see how to use the Laplace transform to solve problems in five different topics in physics and engineering. Those problems involve differential equations, so the first section of this chapter () provides an introduction to the application of the Laplace transform to ordinary and partial differential equations. Once you have an understanding of the general concept of solving a differential equation by applying an integral transform, you can work through specific applications including mechanical oscillations (), electrical circuits (), heat flow (), waves (), and transmission lines (). Each of these applications has been chosen to illustrate a different aspect of using the Laplace transform to solve differential equations, so you may find them useful even if you have little interest in the specific subject matter. And as in every chapter, the final section () of this chapter has a set of problems you can use to check your understanding of the concepts and mathematical techniques presented in this chapter.

Becoming familiar with the Laplace transform F(s) of basic time-domain functions f(t) such as exponentials, sinusoids, powers of t, and hyperbolic functions can be immensely useful in a variety of applications. That is because many of the more complicated functions that describe the behavior of real-world systems and that appear in differential equations can be synthesized as a mixture of these basic functions. And although there are dozens of books and websites that show you how to find the Laplace transform of such functions, much harder to find are explanations that help you achieve an intuitive understanding of why F(s) takes the form it does, that is, an understanding that goes beyond “That’s what the integral gives”. So the goal of this chapter is not just to show you the Laplace transforms of some basic functions, but to provide explanations that will help you see why those transforms make sense.

The Laplace transform is a mathematical operation that converts a function from one domain to another. And why would you want to do that? As you’ll see in this chapter, changing domains can be immensely helpful in extracting information from the mathematical functions and equations that describe the behavior of natural phenomena as well as mechanical and electrical systems. Specifically, when the Laplace transform operates on a function f(t) that depends on the parameter t, the result of the operation is a function F(s) that depends on the parameter s. You’ll learn the meaning of those parameters as well as the details of the mathematical operation that is defined as the Laplace transform in this chapter, and you’ll see why the Fourier transform can be considered to be a special case of the Laplace transform.

The value of knowing the Laplace transforms of the basic functions described in the previous chapter is greatly enhanced by certain properties of the Laplace transform. That is because these properties allow you to determine the transform of much more complicated time-domain functions by combining and modifying the transforms of simple functions such as those discussed in .