1550078481-Ordinary_Differential_Equations__Roberts_

The Laplace Transform Method 231

Not all functions have a Laplace transform as the following example shows.

EXAMPLE 8 A Function with No Laplace Transform, ex

2

2 fCX> 2

Show that .C[ex J = Jo ex e-sx dx does not exist.

SOLUTION

Clearly the integral does not exist if s ::; 0, since the integrand is positive.

Suppose the integral does exist for some s > 0. Then

(7) .C[ex2] = {°'° ex2 e-sx dx = r2s ex(x-s) dx + r = ex(x-s) dx.

lo lo 12 s

The first integral on the right-hand side of equation (7) is positive, since the

integrand is positive for all real x and s. For x ;::: 2s, we have x - s ;::: s and

ex(x-s) ;::: esx. Therefore the second integral on the right-hand side of (7)

satisfies the inequality

r = ex(x-s) dx ;::: 1 = esx dx = 00.

12 s 2s

Thus, for any s > 0, we have

1

.C[ex^2 ] = 2s ex(x-s) dx + 1= ex(x-s) dx ;:::^1 = esx dx = 00.

0 2s 2s

Consequently, .C[ex

2

] does not exist.

For any fixed positive value of s, the factor e-sx, which appears in the

integrand of the definition of the Laplace transform, is a "damping factor"-

a factor which decreases to zero as x increases. Provided the function f(x)
does not "grow too rapidly" as x increases, we expect the defining integral to
converge and, therefore, the Laplace transform to exist. Classes of functions
which do not "grow too rapidly" are said to be of exponential order a, or
simply of exponential order. Such functions are defined as follows.

DEFINITION Exponential Order

A function f(x) is of exponential order a as x --> +oo if and only if

there exist positive constants M and x 0 and a constant a such that

lf(x)I < Meax for x;::: Xo.