9.7: Chapter 7 Answers

Section 7.1

1. \(\frac{1}{s^{2}}\)

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

3. \(\frac{b}{s^{2}-b^{2}}\)

4. \(\frac{-2s+5}{(s-1)(s-2)}\)

5. \(\frac{2}{s^{3}}\)

6. \(\frac{2}{s(s^{2}+4)}\)

7. \(\frac{s^2+8}{s(s^{2}+16)}\)

8. \(\frac{1}{(s^{2}+4)}\)

9. \(\frac{5s}{(s^{2}+4)(s^{2}+9)}\)

10. \(\frac{s^{3}+2s^{2}+4s+32}{(s^{2}+4)(s^{2}+16)}\)

11. \(f(3-)=-1,\: f(3)=f(3+)=1\)

12. \(f(1-)=3,\: f(1)=4,\: f(1+)=1\)

13. \(f\left(\frac{\pi }{2}-\right) =1,\: f\left(\frac{\pi }{2} \right) = f\left(\frac{\pi }{2}+ \right)=2,\: f(\pi -)=0,\: f(\pi )=f(\pi +)=-1\)

14. \(f(1−) = 1,\: f(1) = 2,\: f(1+) = 1,\: f(2−) = 0,\: f(2) = 3,\: f(2+) = 6\)

15. \(\frac{1-e^{-(s+1)}}{s+1}+\frac{e^{-(s+2)}}{s+2}\)

16. \(\frac{1}{s}+e^{-4s}\left(\frac{1}{s^{2}}+\frac{3}{s}\right)\)

17. \(\frac{1-e^{-s}}{s^{2}}\)

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

Section 7.2

1. \({1\over 2}t^2\)

2. \({1\over 6}t^3\)

3. \(t-2t^4\)

4. \(4t-{2\over 3}t^3+{1\over 120}t^5\)

5. \(1+3t+{3\over 2}t^2+{1\over 6}t^3\)

6. \(1+4t+2t^2\)

7. \(t-1+e^{2t}\)

8. \(4+{1\over 4}t^4-e^{-8t}\)

9. \({1\over 4}e^{-t/4}\)

10. \({1\over 5}e^{2t/5}\)

11. \({5\over 7}\sin 7t\)

12. \(10\cos 4t\)

13. \(\cos {t\over 2}\)

14. \({1\over 2}\sin {t\over 2}\)

15. \(2\cos 3t-2\sin 3t\)

16. \(\cos \sqrt 2t+{\sqrt2\over 2}\sin \sqrt2t\)

17. \({1\over 3}-{1\over 3}e^{-3t}\)

18. \(-{1\over 4}+{5\over 4}e^{4t}\)

19. \({1\over 4}e^t+{3\over 4}e^{-3t}\)

20. \({1\over 9}e^{4t}-{1\over 9}e^{-5t}\)

21. \(0.3e^{0.1t}+0.6e^{-0.2t}\)

22. \(\cosh \sqrt3 t-\sqrt3\sinh \sqrt3t\)

23. \({1\over 2}e^{2t}-e^{3t}+{1\over 2}e^{6t}\)

24. \({1\over 2}-e^t-{1\over 3}e^{-t}+{5\over 6}e^{2t}\)

25. \({1\over 5}-{1\over 5}\cos \sqrt5t\)

26. \({1\over 4}\cos 2t+{1\over 4}\sin 2t-{1\over 4}e^{-2t}\)

27. \(-4+3e^{-t}+\cos t+3\sin t\)

28. \({1\over 6\sqrt3}\sinh \sqrt3t-{1\over 6\sqrt3}\sin \sqrt3t\)

29. \({1\over 4}\cos 2t+{1\over 4}\sin 2t-{1\over 4}e^{-2t}\)

30. \(2\cos t+\sin t-2\cos 2t-{1\over 2}\sin 2t\)

31. \(\frac{2}{3}\sin 3t\)

32. \(2\cos 3t+\frac{1}{3}\sin 3t\)

33. \(\cosh 3t+\frac{1}{3}\sinh 3t\)

34. \(1-\cos t\)

35. \(3\cosh t+4\sinh t\)

36. \(3e^{t}+4\cos 3t+\frac{1}{3}\sin 3t\)

37. \(\frac{1}{4}e^{2t}-\frac{1}{4}e^{-2t}-e^{-t}\)

38. \(\frac{1}{5}e^{-4t}-\frac{41}{5}e^{t}+5e^{3t}\)

39. \(-\frac{1}{2}e^{2t}-\frac{13}{10}e^{-2t}-\frac{1}{5}e^{3t}\)

40. \(-\frac{2}{5}e^{-4t}-\frac{3}{5}e^{t}\)

41. \(\frac{3}{20}e^{2t}-\frac{37}{12}e^{-2t}+\frac{1}{3}e^{t}+\frac{8}{5}e^{-3t}\)

42. \(\frac{39}{10}e^{t}+\frac{3}{14}e^{3t}+\frac{23}{105}e^{-4t}-\frac{7}{3}e^{2t}\)

43. \(\frac{4}{5}e^{-2t}-\frac{1}{2}e^{-t}-\frac{3}{10}\cos t+\frac{11}{10}\sin t\)

44. \(\frac{3}{5}\cos 2t+\frac{1}{5}\sin 2t-\frac{3}{5}\cos 3t-\frac{2}{15}\sin 3t\)

45. \(-\frac{4}{15}\cos t+\frac{1}{15}\sin t+\frac{4}{15}\cos 4t-\frac{1}{60}\sin 4t\)

46. \(\frac{5}{3}\cos t+\sin t-\frac{5}{3}\cos 2t-\frac{1}{2}\sin 2t\)

47. \(-\frac{1}{3}\cos \frac{t}{2}+\frac{2}{3}\sin\frac{t}{2}+\frac{1}{3}\cos t-\frac{1}{3}\sin t\)

48. \(\frac{1}{15}\cos\frac{t}{4}-\frac{8}{15}\sin\frac{t}{4}-\frac{1}{15}\cos 4t+\frac{1}{30}\sin 4t\)

49. \(\frac{2}{5}\cos\frac{t}{3}-\frac{3}{5}\sin\frac{t}{3}-\frac{2}{5}\cos\frac{t}{2}+\frac{2}{5}\sin\frac{t}{2}\)

50. \(1-\cos t\)

51. \(-\frac{3}{10}\sin t+\frac{2}{5}\cos t-\frac{3}{4}e^{t}+\frac{7}{20}e^{3t}\)

52. \(-\frac{1}{2}e^{t}+\frac{7}{10}e^{-t}-\frac{1}{5}\cos 2t+\frac{3}{5}\sin 2t\)

Section 7.3

1. \(y=-1+e^t\)

2. \(y=-3e^{-t/2}\)

3. \(y={1\over 10}e^{4t}+{19\over 10}e^{-6t}\)

4. \(y={1\over 13}e^{t}-{1\over 13}\cos 5t+{5\over 13}\sin 5t\)

5. \(y={4\over 3}e^{-t}-{1\over 3}e^{-4t}\)

6. \(y={5\over 2}-2e^{3t}-{3\over 5}e^{-t}+{11\over 10}e^{4t}\)

7. \(y=10\cos t+2\sin t-\sqrt2\sin\sqrt2 t\)

8. \(y={1\over 10}e^{t}-{1\over 30}\sin 3t-{1\over 10}\cos 3t\)

9. \(y={1\over 2}e^{-t}+{5\over 18}e^t-{8\over 9}e^{-t/2}+{1\over 9}e^{-2t}\)

10. \(y={13\over 60}e^{t}-{13\over 20}e^{-t}+{16\over 39}e^{-2t}+{3\over 130}\cos 3t-{1\over 65}\sin 3t\)

11. \(y=\frac{1}{6}e^{t}-\frac{9}{2}e^{-t}+\frac{16}{3}e^{-2t}\)

12. \(y=-\frac{1}{3}+\frac{8}{15}e^{3t}+\frac{4}{5}e^{-2t}\)

13. \(y=-\frac{23}{15}e^{-2t}+\frac{1}{3}e^{t}+\frac{1}{5}e^{3t}\)

14. \(y=-\frac{1}{4}e^{2t}+\frac{17}{20}e^{-2t}+\frac{2}{5}e^{3t}\)

15. \(y=\frac{11}{15}e^{-2t}+\frac{1}{6}e^{t}+\frac{1}{10}e^{3t}\)

16. \(y=e^{t}+2e^{-2t}-2e^{-t}\)

17. \(y=\frac{5}{3}\sin t-\frac{1}{3}\sin 2t\)

18. \(y=4e^{t}-4e^{2t}+e^{3t}\)

19. \(y=-\frac{7}{2}e^{2t}+\frac{13}{3}e^{t}+\frac{1}{6}e^{4t}\)

20. \(y=\frac{5}{2}e^{t}-4e^{2t}+\frac{1}{2}e^{3t}\)

21. \(y=\frac{1}{3}e^{t}-2e^{-t}+\frac{5}{3}e^{-2t}\)

22. \(y=2-e^{-2t}+e^{t}\)

23. \(y=1-\cos 2t+\frac{1}{2}\sin 2t\)

24. \(y=-\frac{1}{3}+\frac{8}{15}e^{3t}+\frac{4}{5}e^{-2t}\)

25. \(y=\frac{1}{6}e^{t}-\frac{2}{3}e^{-2t}+\frac{1}{2}e^{-t}\)

26. \(y=-1+e^{t}+e^{-t}\)

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

28. \(y=\frac{7}{3}-\frac{7}{2}e^{-t}+\frac{1}{6}e^{3t}\)

29. \(y=1+\cos t\)

30. \(y=t+\sin t\)

31. \(y=t-6\sin t+\cos t+\sin 2t\)

32. \(y=e^{-t}+4e^{-2t}-4e^{-3t}\)

33. \(y=-\sin t-2\cos t+3e^{3t}+e^{-t}\)

34. \(y=\frac{1}{5}e^{2t}-\frac{4}{3}e^{t}+\frac{32}{15}e^{-t/2}\)

35. \(y=\frac{1}{7}e^{2t}-\frac{2}{5}e^{t/2}+\frac{9}{35}e^{-t/3}\)

Section 7.4

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

2. \({6\over (s+2)^4}\)

3. \({1\over s-6}+{3\over (s-6)^2}-{2\over (s-6)^3}\)

4. \(\frac{t^{3}e^{7t}}{2}\)

5. \(t^{2}e^{7t}+\frac{17}{6}t^{3}e^{7t}\)

6. \(e^{2t}\left(\frac{1}{6}t^{3}+\frac{1}{6}t^{4}+\frac{1}{40}t^{5}\right)\)

7. \((1-t)e^{-t}\)

8. \(\left( 1-t-t^{2}-\frac{1}{6}t^{3}\right)e^{-t}\)

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

10. \(y=e^{-t}+2te^{-t}\)

11. \(y={1\over 9}t+{2\over 27}-{2\over 27}e^{3t}+{10\over 9}te^{3t}\)

12. \(y=-3\cos t-2\sin t+e^{-t}(2+5t)\)

13. \(y=\frac{e^{-t/2}}{10}(5t+26)-\frac{1}{5}(3\cos t+\sin t)\)

14.  \(T(t)=105e^{-0.425t}+78\) is the temperature function. It takes about \(0.389\) minutes for the coffee to reach \(167\) degrees.

15. \(x(t)=-5\cos(2t)+5\cos(5t)\)

Section 7.5

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