Page 255 - Schaum's Outline of Differential Equations

238 CONVOLUTIONS AND THE UNIT STEP FUNCTION [CHAP. 23

Fig. 23-6 Fig. 23-7

23.13. Find £{g(x)} if g(x) = •

2
2
If we define/(X) = x , then g(x) can be given compactly as g(x) = u(x - 4)f(x - 4) = u(x - 4)(x - 4) . Then,
noting that ££{/(X)} = F(s) = 2/s 3 and using Theorem 23.4, we conclude that

23.14. Find £{g(x)} if g(x) =

2
We first determine a function f(x) such that/(jc - 4) =x . Once this has been done, g(x) can be written as
g (x) = u (x — 4)/(x — 4) and Theorem 23.4 can be applied. Now, f(x - 4) = x 2 only if

Since

it follows that

23.15. Prove Theorem 23.1.

Making the substitution r = x — t in the right-hand side of Eq. (23.1), we have