A Student's Guide to Laplace Transforms

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.