Page 230 - Schaum's Outline of Differential Equations

CHAP. 21] THE LAPLACE TRANSFORM 213

hence the integral diverges. For s ^ 0,

when s < 0, —sR > 0; hence the limit is °° and the integral diverges. When s > 0, —sR < 0; hence, the limit is 1/5 and
the integral converges.

21.4. Find the Laplace transform off(x) = 1.
Using Eq. (21.1) and the results of Problem 21.3, we have

(See also entry 1 in Appendix A.)

1
21.5. Find the Laplace transform of f(x) = x .
Using Eq. (21.1) and integration by parts twice, we find that

2
For s < 0, lim/;^^ [- (R ls)e sR ] = °°, and the improper integral diverge. For s > 0, it follows from repeated use of
L'Hopital's rule that

3
3
Also, lim/^co [- (2/s )e sR ] = 0 directly; hence the integral converges, and F(s) = 2/s . For the special cases s = 0, we have

2
3
Finally, combining all cases, we obtain ££{jc } = 21s , s > 0. (See also entry 3 in Appendix A.)
21.6. Find
Using Eq. (21.1), we obtain

Note that when s < a, the improper integral diverges. (See also entry 7 in Appendix A.)