35: Section 9.5 Answers
- Page ID
- 103655
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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. \(1+ {\cal U} (t-4)(t-1);\quad\frac{1}{s}+e^{-4s}\left(\frac{1}{s^{2}}+\frac{3}{s}\right)\)
2. \(t+ {\cal U} (t-1)(1-t);\quad\frac{1-e^{-s}}{s^{2}}\)
3. \(2t-1- {\cal U} (t-2)(t-1);\quad\left(\frac{2}{s^{2}}-\frac{1}{s}\right)-e^{-2s}\left(\frac{1}{s^{2}}+\frac{1}{s}\right)\)
4. \(1+ {\cal U} (t-1)(t+1);\quad\frac{1}{s}+e^{-s}\left(\frac{1}{s^{2}}+\frac{2}{s}\right)\)
5. \(t-1+ {\cal U} (t-2)(5-t);\quad\frac{1}{s^{2}}-\frac{1}{s}-e^{-2s}\left(\frac{1}{s^{2}}-\frac{3}{s}\right)\)
6. \(t^{2}(1- {\cal U} (t-1));\quad\frac{2}{s^{3}}-e^{-s}\left(\frac{2}{s^{3}}+\frac{2}{s^{2}}+\frac{1}{s}\right)\)
7. \( {\cal U} (t-2)(t^{2}+3t);\quad e^{-2s}\left(\frac{2}{s^{3}}+\frac{7}{s^{2}}+\frac{10}{s}\right)\)
8. \(t^{2}+2+ {\cal U} (t-1)(t-t^{2}-2);\quad\frac{2}{s^{3}}+\frac{2}{s}-e^{-s}\left(\frac{2}{s^{3}}+\frac{1}{s^{2}}+\frac{2}{s}\right)\)
9. \(te^{t}+ {\cal U} (t-1)(e^{t}-te^{t});\quad\frac{1-e^{-(s-1)}}{(s-1)^{2}}\)
10. \(e^{-t}+ {\cal U} (t-1)(e^{-2t}-e^{-t});\quad\frac{1-e^{-(s+1)}}{s+1}+\frac{e^{-(s+2)}}{s+2}\)
11. \(-t+2 {\cal U} (t-2)(t-2)- {\cal U} (t-3)(t-5);\quad-\frac{1}{s^{2}}+\frac{2e^{-2s}}{s^{2}}+e^{-3s}\left(\frac{2}{s}-\frac{1}{s^{2}}\right)\)
12. \(\left[ {\cal U} (t-1)- {\cal U} (t-2)\right] t;\quad e^{-s}\left(\frac{1}{s^{2}}+\frac{1}{s}\right)-e^{-2s}\left(\frac{1}{s^{2}}+\frac{2}{s}\right)\)
13. \(t+ {\cal U} (t-1)(t^{2}-t)- {\cal U} (t-2)t^{2};\quad\frac{1}{s^{2}}+e^{-s}\left(\frac{2}{s^{3}}+\frac{1}{s^{2}}\right)-e^{-2s}\left(\frac{2}{s^{3}}+\frac{4}{s^{2}}+\frac{4}{s}\right)\)
14. \(t+ {\cal U} (t-1)(2-2t)+ {\cal U} (t-2)(4+t);\quad\frac{1}{s^{2}}-2\frac{e^{-s}}{s^{2}}+e^{-2s}\left(\frac{1}{s^{2}}+\frac{6}{s}\right)\)
15. \(\sin t+ {\cal U} (t-\pi /2)\sin t+ {\cal U} (t-\pi )(\cos t-2\sin t);\quad \frac{1+e^{-\frac{\pi }{2}s}s-e^{-\pi s}(s-2)}{s^{2}+1}\)
16. \(2-2 {\cal U} (t-1)t+ {\cal U} (t-3)(5t-2);\quad\frac{2}{s}-e^{-s}\left(\frac{2}{s^{2}}+\frac{2}{s}\right)+e^{-3s}\left(\frac{5}{s^{2}}+\frac{13}{s}\right)\)
17. \(3+ {\cal U} (t-2)(3t-1)+ {\cal U} (t-4)(t-2);\quad\frac{3}{s}+e^{-2s}\left(\frac{3}{s^{2}}+\frac{5}{s}\right)+e^{-4s}\left(\frac{1}{s^{2}}+\frac{2}{s}\right)\)
18. \((t+1)^{2}+ {\cal U} (t-1)(2t+3);\quad\frac{2}{s^{3}}+\frac{2}{s^{2}}+\frac{1}{s}+e^{-s}\left(\frac{2}{s^{2}}+\frac{5}{s}\right)\)
19. \( {\cal U} (t-2)e^{2(t-2)}=\left\{\begin{array}{cc}{0,}&{0\leq t<2,}\\{e^{2(t-2)},}&{t\geq 2}\end{array} \right.\)
20. \( {\cal U} (t-1)\left(1-e^{-(t-1)}\right)=\left\{\begin{array}{cc}{0,}&{0\leq t<1,}\\{1-e^{-(t-1)},}&{t\geq 1}\end{array} \right.\)
21. \( {\cal U} (t-1)\frac{(t-1)^{2}}{2}+ {\cal U} (t-2)(t-2)=\left\{\begin{array}{cc}{0,}&{0\leq t<1,}\\{\frac{(t-1)^{2}}{2},}&{1\leq t<2,}\\{\frac{t^{2}-3}{2},}&{t\geq 2}\end{array} \right.\)
22. \(2+t+ {\cal U} (t-1)(4-t)+ {\cal U} (t-3)(t-2)=\left\{\begin{array}{cc}{2+t,}&{0\leq t<1,}\\{6,}&{1\leq t<3,}\\{t+4,}&{t\geq 3}\end{array} \right.\)
23. \(5-t+ {\cal U} (t-3)(7t-15)+\frac{3}{2} {\cal U} (t-6)(t-6)^{2}=\left\{\begin{array}{cc}{5-t,}&{0\leq t<3,}\\{6t-10,}&{3\leq t<6,}\\{44-12t+\frac{3}{2}t^{2},}&{t\geq 6}\end{array} \right.\)
24. \( {\cal U} (t-\pi )e^{-2(t-\pi )}(2\cos t-5\sin t)=\left\{\begin{array}{cc}{0,}&{0\leq t<\pi ,}\\{e^{-2(t-\pi )}(2\cos t-5\sin t)}&{t\geq\pi }\end{array} \right.\)
25. \(1-\cos t+ {\cal U} (t-\pi /2)(3\sin t+\cos t)=\left\{\begin{array}{cc}{1-\cos t,}&{0\leq t<\frac{\pi }{2},}\\{1+3\sin t,}&{t\geq\frac{\pi }{2}}\end{array} \right.\)
26. \( {\cal U} (t-2)(4e^{-(t-2)}-4e^{2(t-2)}+2e^{(t-2)})=\left\{\begin{array}{cc}{0,}&{0\leq t<2,}\\{4e^{-(t-2)}-4e^{2(t-2)}+2e^{(t-2)},}&{t\geq 2}\end{array} \right.\)
27. \(1+t+ {\cal U} (t-1)(2t+1)+ {\cal U} (t-3)(3t-5)=\left\{\begin{array}{cc}{t+1,}&{0\leq t<1,}\\{3t+2,}&{1\leq t<3,}\\{6t-3,}&{t\geq 3}\end{array} \right.\)
28. \(1-t^{2}+ {\cal U} (t-2)\left(-\frac{t^{2}}{2}+2t+1\right)+ {\cal U} (t-4)(t-4)=\left\{\begin{array}{cc}{1-t^{2},}&{0\leq t<2,}\\{-\frac{3t^{2}}{2}+2t+2,}&{2\leq t<4,}\\{-\frac{3t^{2}}{2}+3t-2,}&{t\geq 4}\end{array} \right.\)
29. \(y=3(1-\cos t)-3 {\cal U} (t-\pi )(1+\cos t)\)
30. \(y=3-2\cos t+2 {\cal U} (t-4)(t-4-\sin (t-4))\)
31. \(y=-\frac{15}{2}+\frac{3}{2}e^{2t}-2t+\frac{ {\cal U} (t-1)}{2}(e^{2(t-1)}-2t+1)\)
32. \(y=\frac{1}{2}e^{t}+\frac{13}{6}e^{-t}+\frac{1}{3}e^{2t}+ {\cal U} (t-2)\left(-1+\frac{1}{2}e^{t-2}+\frac{1}{2}e^{-(t-2)}+\frac{1}{2}e^{t+2}-\frac{1}{6}e^{-(t-6)}-\frac{1}{3}e^{2t}\right)\)
33. \(y=-7e^{t}+4e^{2t}+ {\cal U} (t-1)\left(\frac{1}{2}-e^{t-1}+\frac{1}{2}e^{2(t-1)}\right)-2 {\cal U} (t-2)\left(\frac{1}{2}-e^{t-2}+\frac{1}{2}e^{2(t-2)}\right)\)
34. \(y=\frac{1}{3}\sin 2t-3\cos 2t+\frac{1}{3}\sin t-2 {\cal U} (t-\pi )\left(\frac{1}{3}\sin t+\frac{1}{6}\sin 2t\right)+ {\cal U} (t-2\pi )\left(\frac{1}{3}\sin t-\frac{1}{6}\sin 2t\right)\)
35. \(y=\frac{1}{4}-\frac{31}{12}e^{4t}+\frac{16}{3}e^{t}+ {\cal U} (t-1)\left(\frac{2}{3}e^{t-1}-\frac{1}{6}e^{4(t-1)}-\frac{1}{2}\right)+ {\cal U} (t-2)\left(\frac{1}{4}+\frac{1}{12}e^{4(t-2)}-\frac{1}{3}e^{t-2}\right)\)
36. \(y=\frac{1}{8}(\cos t-\cos 3t)+\frac{1}{8} {\cal U} \left(t-\frac{3\pi }{2}\right)\left(\sin t-\cos t+\sin 3t-\frac{1}{3}\cos 3t\right)\)
37. \(y=\frac{t}{4}-\frac{1}{8}\sin 2t+\frac{1}{8} {\cal U} \left(t-\frac{\pi }{2}\right)(\pi\cos 2t-\sin 2t+2\pi -2t)\)
38. \(y=t-\sin t-2 {\cal U} (t-\pi )(t+\sin t+\pi\cos t)\)
39. \(y= {\cal U} (t-2)\left(t-\frac{1}{2}+\frac{e^{2(t-2)}}{2}-2e^{t-2}\right)\)
40. \(y=t+\sin t+\cos t- {\cal U} (t-2\pi )(3t-3\sin t-6\pi\cos t)\)
41. \(y=\frac{1}{2}+\frac{1}{2}e^{-2t}-e^{-t}+ {\cal U} (t-2)\left(2e^{-(t-2)}-e^{-2(t-2)}-1\right)\)
42. \(y=-\frac{1}{3}-\frac{1}{6}e^{3t}+\frac{1}{2}e^{t}+ {\cal U} (t-1)\left(\frac{2}{3}+\frac{1}{3}e^{3(t-1)}-e^{t-1}\right)\)
43. \(y=\frac{1}{4}\left(e^{t}+e^{-t}(11+6t)\right)+ {\cal U} (t-1)(te^{-(t-1)}-1)\)
44. \(y=e^{t}-e^{-t}-2te^{-t}- {\cal U} (t-1)\left(e^{t}-e^{-(t-2)}-2(t-1)e^{-(t-2)}\right)\)
45. \(y=te^{-t}+e^{-2t}+ {\cal U} (t-1)\left(e^{-t}(2-t)-e^{-(2t-1)}\right)\)
46. \(y=\frac{t^{2}e^{2t}}{2}-te^{2t}- {\cal U} (t-2)(t-2)^{2}e^{2t}\)
47. \(y=\frac{t^{4}}{12}+1-\frac{1}{12} {\cal U} (t-1)(t^{4}+2t^{3}-10t+7)+\frac{1}{6} {\cal U} (t-2)(2t^{3}+3t^{2}-36t+44)\)
48. \(y=\frac{1}{2}e^{-t}(3\cos t+\sin t)+\frac{1}{2}- {\cal U} (t-2\pi )\left(e^{-(t-2\pi )}\left((\pi -1)\cos t+\frac{2\pi -1}{2}\sin t\right)+1-\frac{t}{2}\right)-\frac{1}{2} {\cal U} (t-3\pi )(e^{-(t-3\pi )}(3\pi\cos t+(3\pi +1)\sin t)+t)\)