In problem use the Laplace transform to solve the given initial-value problem. Use the table of Laplace transforms in Appendix III as needed.

y'' + y = sin t, y(0) = 1, y'(0) = -1

Appendix III

-1 7. sin kt 8. cos kt s + k 2k2 9. sin? kt s(s + 4k?) 2 + 2k2 10. cos? kt s(s? + 4k2) 11." class="fr-fic fr-dii">

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= F(s) 1. 1 2. t 3. 1" n! na positive integer 4. 1/2 5. 12 2532 I(a + 1) 6. 1a a >-1 7. sin kt 8. cos kt s + k 2k2 9. sin? kt s(s + 4k?) 2 + 2k2 10. cos? kt s(s? + 4k2) 11. е S - a 12. sinh kt 13. cosh kt s2 - k2 212 14. sinh?kt s(s? – 4k2) 52 – 2k? 15. cosh kt s( - 4k) 16. teat (s - a) n! 17. "eat n a positive integer (s - ay+1 -