Page 253 - Schaum's Outline of Differential Equations

236 CONVOLUTIONS AND THE UNIT STEP FUNCTION [CHAP. 23

x
Defining F(s) = ll(s — 1) and G(s) = ll(s + 1), we have from Appendix A that/(jc) = e" and g(x) = e . It follows from
Eq. (23.2) that

23.6. Find 5T 1 by convolutions.
Note that

Defining F(s) = Us and G(s) = l/(s 2 + 4), we have from Appendix A that/(jc) = 1 and g(x) = jsin2x . It now follows
from Eq. (23.2) that

See also Problem 22.19.

23.7. Find $~ l by convolutions.

If we define F(s) = G(s) = l/(s - 1), then/(jc) =g(x) = e"and

23.8. Use the definition of the Laplace transform to find !£{u(x - c)} and thereby prove Theorem 23.3.
It follows directly from Eq. (21.1) that