Page 237 - Schaum's Outline of Differential Equations

220 THE LAPLACE TRANSFORM [CHAP. 21

21.23. Find the Laplace transform of the function graphed in Fig. 21-3.
Note that f(x) is periodic with period K>=2n, and in the interval 0 < x < 2nit can be defined analytically by

From Eq. (21.8), we have

Since

it follows that

21.24. Find

Using Eq. (21.4) with a = - 4 on the results of Problem 21.19, we obtain

It now follows from Eq. (21.7) that

and then from Property 21.3 with n = 1,

Finally, using Eq. (21.4) with a = 4, we conclude that the required transform is

21.25. Find the Laplace transforms at (a) t, (b) e^, and (c) sin at, where a denotes a constant.
Using entries 2, 7, and 8 of Appendix A with x replaced by t, we find the Laplace transforms to be,