Page 244 - Schaum's Outline of Differential Equations

CHAP. 22] INVERSE LAPLACE TRANSFORMS 227

22.8. Find

No function of this form appears in Appendix A. But, by completing the square, we obtain

2
2
s - 2s + 9 = (s - 2s + 1) + (9 -1) = (s - I) 2
Hence,

Then, using Property 22.1 and entry 15 of Appendix A with a = and b = 1, we find that

22.9. Find

No function of this form appears in Appendix A. Completing the square in the denominator, we have
2
2
2
s + 4s + 8 = (s + 4s + 4) + (8 - 4) = (s + 2) + (2) 2
Hence,
This expression also is not found in Appendix A. However, if we rewrite the numerator as s + 4 = (s + 2) + 2 and
then decompose the fraction, we have

Then, from entries 15 and 16 of Appendix A,

22.10. Find

No function of this form appears in Appendix A. Completing the square in the denominator, we obtain

so that

We now rewrite the numerator as

so that