Nine - Laplace Transforms

Abstract

This chapter discusses the concept of Laplace transforms. In applying the method of Laplace transforms to the initial value problem, three main steps are undertaken. The first step transforms a hard problem into a relatively easy problem. Then the easy problem is solved by finding X. Finally, X is inverted; that is, a solution x of the original problem is found from the solution of the transformed problem. This same procedure is followed in the solution of more complicated initial value problems. The chapter discusses the applications of Laplace transforms to differential equations. In establishing the existence of the Laplace transform of a function, it is necessary to show that a certain improper integral converges. The theory of Laplace transforms is applied to the solution of initial value problems. The chhapter discusses problems where the differential equation, or system of differential equations, is linear and has constant coefficients.