Page 252 - Schaum's Outline of Differential Equations

CHAP. 23] CONVOLUTIONS AND THE UNIT STEP FUNCTION 235

Solved problems

2x
23.1. Find/(;t) * g (x) \vhenf(x) = (P x and g (x) = e .
3
Here/(0 = e ', g(x-t) = e 2(x -'\ and

23.2. Find g(x) *f(x) for the two functions in problem 23.1 and verify Theorem 23.1.
2t
0
3
With/(;t - 0 = e ^ and g(t) = e ,

which, from Problem 23.1 equals f(x) * g(x).

2
23.3. Find f(x) * g(x) when f(x) = x and g(x) = x .
2
2
Here/(0 = t and g(x - t) = (x - I) 2 = x - 2xt + t . Thus,

23.4. Find by convolutions.

Note that

3
2
Defining F(s) = ll(s — 3) and G(s) = ll(s — 2), we have from Appendix A that/(jc) = e * and g(x) = e *. It follows from
Eq. (23.2) and the results of Problem 23.1 that

23.5. Find 5T 1 by convolutions.

Note that