1550078481-Ordinary_Differential_Equations__Roberts_

The Laplace Trans] orm Method

EXAMPLE 1 Using the Convolution Theorem

to Find an Inverse Laplace Transform

1

Find a function h(x) whose Laplace transform is H(s) = 2 ( ).

s s + 1

SOLUTION

We can rewrite H(s) as the product

1 1

H(s) = 2 - = F(s)G(s),

s s + 1

253

1 1

where F(s) = 82 = .C[x] and G(s) =

8

+

1

= .C[e-x]. By the convolution

theorem and the commutative property of the convolution operator, we have

and

h(x) = e-x * x = 1 x e-(x-t;)~ d~.

Evaluating the second integral, which is slightly simpler, we find

The next example shows how to solve example 1 using partial fraction
expansion.

EXAMPLE 2 Using Partial Fraction Expansion

to Find an Inverse Laplace Transform

Use partial fraction expansion to find a function h(x) whose Laplace trans-

form is

H(s) - ~-

1

----.,...

• s^2 (s + lf