Introduction

In many problems, a function $f(t), \;\; t \in [a, \; b]$ is transformed to another function $F(s)$ through a relation of the type:

$$F(s) = \int_a^b K(t,s) f(t) dt$$

where $K(t,s)$ is a known function. Here, $F(s)$ is called integral transform of $f(t)$ . Thus, an integral transform sends a given function $f(t)$ into another function $F(s)$ . This transformation of $f(t)$ into $F(s)$ provides a method to tackle a problem more readily. In some cases, it affords solutions to otherwise difficult problems. In view of this, the integral transforms find numerous applications in engineering problems. Laplace transform is a particular case of integral transform (where $f(t)$ is defined on $[0, \infty)$ and $K(s,t) = e^{-st}$ ). As we will see in the following, application of Laplace transform reduces a linear differential equation with constant coefficients to an algebraic equation, which can be solved by algebraic methods. Thus, it provides a powerful tool to solve differential equations.

It is important to note here that there is some sort of analogy with what we had learnt during the study of logarithms in school. That is, to multiply two numbers, we first calculate their logarithms, add them and then use the table of antilogarithm to get back the original product. In a similar way, we first transform the problem that was posed as a function of $f(t)$ to a problem in $F(s)$ , make some calculations and then use the table of inverse Laplace transform to get the solution of the actual problem.

In this chapter, we shall see same properties of Laplace transform and its applications in solving differential equations.