Chapter 5
The Laplace Transform Method
In sections 4.3 and 4.4 we showed how to solve homogeneous and nonhomo-
geneous linear differential equations with constant coefficients and in section
4.5 we showed how to solve initial value problems in which the differential
equations were homogeneous or nonhomogeneous linear differential equations
with constant coefficients. The technique consisted of finding the general solu-
tion of the differential equation and then choosing the constants in the general
solution to satisfy the specified initial conditions. In this chapter, we present
the Laplace transform method for solving homogeneous and nonhomogeneous
linear differential equations with constant coefficients and their corresponding
initial value problems. We begin by examining the Laplace transform and its
properties.
5.1 The Laplace Transform and Its Properties
The Laplace transform is named in honor of the French mathematician and
astronomer Pierre Simon Marquis de Laplace (1749-1827). Laplace studied
the integral appearing in the definition of the transform in 1779 in conjunction
with his research on probability. However, most of the results and techniques
presented in this chapter were developed by the English electrical engineer
Oliver Heaviside (1850-1925) more than a century later. The Laplace trans-
form is defined as follows.
DEFINITION Laplace Transform
Let f(x) be a function defined on the interval [O, +oo). The Laplacetransform of f(x) is
L'.[f(x)] = fo
00f(x)e-sx dx = F(s)
provided the improper integral exists for s sufficiently large.223