36: Section 9.6 Answers
- Page ID
- 103656
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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. \({1\over s^2+1}{1\over 1-e^{-\pi s}}\)