36: Section 9.6 Answers

1. \({s^2-4\over (s^2+4)^2}\)

2. \({6s^2+2\over (s^2-1)^3}\)

3. \({12s-24\over ((s-2)^2+36)^2}\)

4. \({6s\over (s^2-9)^2}\)

5. \({2s(s^2-3)\over (s^2+1)^3}\)

6. \({(s+3)^2-9\over ((s+3)^2+9)^2}\)

7. \({1\over s(s-1)}\)

8. \({s+1\over s[(s+1)^2+1]}\)

9. \({1\over s^2(s-1)}\)

10. \({3s^2+1\over s^2(s^2+1)^2}\)

11. \({1\over s^2+1}\)

12. \({2\over (s^2+1)^2}\)

13. \({s\over (s^2+1)^2}\)

14. \({3s+1\over s^2(s+1)^3}\)

15. \(\frac{as}{(s^{2}+a^{2})(s^{2}+b^{2})}\)

16. \(\frac{a}{(s-1)(s^{2}+a^{2})}\)

17. \(\frac{as}{(s^{2}-a^{2})^{2}}\)

18. \(\frac{(s-1)\omega }{((s-1)^{2}+\omega ^{2})^{2}}\)

19. \(\frac{2}{(s-2)^{3}(s-1)^{2}}\)

20. \(\frac{s+1}{(s+2)^{2}\left[(s+1)^{2}+\omega ^{2}\right]}\)

21. \(\frac{1}{(s-3)((s-1)^{2}-1)}\)

22. \(\frac{2}{(s-2)^{2}(s^{2}+4)}\)

23. \(\frac{6}{s^{4}(s-1)}\)

24. \(\frac{3\cdot 6!}{s^{7}\left[(s+1)^{2}+9\right]}\)

25. \(\frac{12}{s^{7}}\)

26. \(\frac{2\cdot 7!}{s^{8}\left[ (s+1)^{2}+4\right]}\)

27. \(\frac{48}{s^{5}(s^{2}+4)}\)

28. \(2t\sin t\)

29. \(t\cos t\)

30. \(\frac{2te^{2t}}{3}\sin 9t\)

31. \(e^{2t}t\cos t\)

32. \(\frac{1}{2}\int_{0}^{t}u\sin 2(t-u )du \)

33. \(\int_{0}^{t}e^{-2u }\cos 3(t-u )du \)

34. \(\frac{1}{2}\int_{0}^{t}\sin 2u\cos 3(t-u )du\) or \(\frac{1}{3}\int_{0}^{t}\sin 3u\cos 2(t-u )du \)

35. \(\int_{0}^{t}\cos u\sin (t-u )du \)

36. \(\int_{0}^{t}e^{au }du \)

37. \(e^{-t}\int_{0}^{t}\sin (t-u )du \)

38. \(e^{-2t}\int_{0}^{t}u e^{u }\sin (t-u )du \)

39. \(\frac{e^{-2t}}{2}\int_{0}^{t}u ^{2}(t-u )e^{3u }du \)

40. \(\int_{0}^{t}(t-u )e^{u }\cos u du \)

41. \(\int_{0}^{t}e^{-3u }\cos u\cos 2(t-u )du \)

42. \(\frac{1}{4!5!}\int_{0}^{t}u ^{4}(t-u )^{5}e^{3u }du \)

43. \(\frac{1}{4}\int_{0}^{t}u ^{2}e^{u }\sin 2(t-u )du \)

44. \(\frac{1}{2}\int_{0}^{t}u (t-u )^{2}e^{2(t-u )}du \)

45. \(\frac{1}{5!6!}\int_{0}^{t}(t-u )^{5}e^{2(t-u )}u ^{6}du \)

46. \(y=(3t+4)\sin t-(2t+6)\cos t\)

47. \(y=-(2t+2)\cos 2t+\sin 2t+3\cos t\)

48. \(y=-{1\over 2}e^{-t}+{1\over 2}\cos t-{1\over 2}t\cos t+{1\over 2}t\sin t\)

49. \(y=2\cos 3t+{5\over 3}\sin 3t+{1\over 6}t\sin 3t\)

50. \(y={1\over 4}\sin 4t+{1\over 8}t\sin 4t-{1\over 8}(t-\pi)\sin 4(t-\pi){\cal U}(t-\pi)\)

51. \(y=e^{t}\sin t-te^t\cos t\)

52. \(y=\cos t-{1\over 2}\sin t-{1\over 2}t\cos t\)

53. \(y=1-\left(1-\cos (t-\pi/2)\right){\cal U}(t-\pi/2)+{1\over 2}(t-\pi/2)\sin (t-\pi/2){\cal U}(t-\pi/2)\)

54. \(y=\sin t\)

55. \(y={2\over 5}t+{8\over 5\sqrt 5}\sin \sqrt5 t\)

56. \(y={1\over 8}e^t+{3\over 4}te^t+{1\over 4}t^2e^t-{1\over 8}e^{-t}\)

57. \(y=4e^{-t}-7te^{-t}+4t^2e^{-t}\)

58. \(y=e^{-t}\)

59. \(y=\cos t+\sin t\)

60. \(y={1\over 2}\cos 2t+{1\over 4}\sin 2t+{1\over 2}\cosh 2t+{1\over 4}\sinh 2t\)

61. \(y={1\over 2}t-{1\over 12}t^3\)

62. \(y=\sin t-{1\over 2}t\sin t\)

63. \(y=te^{-3t}\)

64. \(y=\sin t\)

65. \(y=te^{-t}\)

66. \(y=1+2te^{t}\)

67. \(y=t+\frac{t^{2}}{2}\)

68. \(y=4+\frac{5}{2}t^{2}+\frac{1}{24}t^{4}\)

69. \(y=1-t\)

70. \({1-e^{-as}\over s(1+e^{-as})}\)

71. \({1\over s(1+e^{-as})}\)

72. \({a\over s}\left({1\over bs}-{1\over e^{bs}-1}\right)\)

73. \({1-e^{-s}\over s^2(1-e^{-2s})}\)

74. \({1\over s^2+1}\coth {\pi s\over 2}\)

75. \({1\over s^2+1}{1\over 1-e^{-\pi s}}\)

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