1550078481-Ordinary_Differential_Equations__Roberts_

252 Ordinary Differential Equations

of integration is shown in Figure 5.2a. Assuming that the order of integration
can b e interchanged, we find

(3) .C[f(x) * g(x)] = 1= [i= f(x - ~)e-sx dx] g(~) d~.

See Figure 5.2b. In the innermost integral in equation (3), we make the change

of variable rJ = x - ~ and thereby obtain

= .C[f(x)].C[g(x)].

Thus, we have proven the following convolution theorem.

CONVOLUTION THEOREM

If .C[f(x)] and .C[g(x) ] both exist for s >a, then

.C[f(x)].C[g(x)] = .C[f(x) * g(x)] for s >a.

di; di;

x dx

x

x dx

x

a. Equation (2) b. Equation (3)

Figure 5.2 Domain and Order of Integration