18.1. INTRODUCTION

A transformation is a mathematical device which converts one function into another. For example, when the differential operator operates on f( x) = sin x, it gives a new function

The Laplace transform or Laplace transformation is widely used by scientists and engineers. It is particularly effective in solving linear differential equations ordinary as well as partial. It reduces an ordinary differential equation into an algebraic equation.

The Laplace transform quickly gives the solution of differential equations with given initial conditions without the necessity of first finding the general solution and then evaluating the arbitrary constants.

18.2. DEFINITION

Let f( t) be a function of t defined for all t ? 0. Then the Laplace transform of f( t), denoted by L{ f( t)}, is defined by

provided that the integral exists, s is a parameter which may be real or complex.

L{ f( t)} is clearly a function of s and is briefly written as f( s) i.e., L{ f( t)} = f( s).

18.3. THE LINEARITY PROPERTY

If c ₁, c ₂ are constants and f, g are functions of t, then

By definition,

The result can easily by generalized.

18.4. LAPLACE TRANSFORMS OF SOME ELEMENTARY FUNCTIONS

1. 

   
   

2. , where n is a positive integer.

   
   

   provided that s > 0 and n + 1 > 0 i.e., n