A.10.6 Section 10.6 Answers
- Page ID
- 43739
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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. \({\bf y}=c_{1}e^{2t}\left[\begin{array}{c}{3\cos t+\sin t}\\[4pt]{5\cos t}\end{array}\right]+c_{2}e^{2t}\left[\begin{array}{c}{3\sin t-\cos t}\\[4pt]{5\sin t}\end{array}\right]\)
2. \({\bf y}=c_{1}e^{-t}\left[\begin{array}{c}{5\cos 2t+\sin 2t}\\[4pt]{13\cos 2t}\end{array}\right]+c_{2}e^{-t}\left[\begin{array}{c}{5\sin 2t-\cos 2t}\\[4pt]{13\sin 2t}\end{array}\right]\)
3. \({\bf y}=c_{1}e^{3t}\left[\begin{array}{c}{\cos 2t+\sin 2t}\\[4pt]{2\cos 2t}\end{array}\right]+c_{2}e^{3t}\left[\begin{array}{c}{\sin 2t-\cos 2t}\\[4pt]{2\sin 2t}\end{array}\right]\)
4. \({\bf y}=c_{1}e^{2t}\left[\begin{array}{c}{\cos 3t-\sin 3t}\\[4pt]{\cos 3t}\end{array}\right]+c_{2}e^{2t}\left[\begin{array}{c}{\sin 3t+\cos 3t}\\[4pt]{\sin 3t}\end{array}\right]\)
5. \({\bf y}=c_{1}\left[\begin{array}{c}{-1}\\[4pt]{-1}\\[4pt]{2}\end{array}\right]e^{-2t}+c_{2}e^{4t}\left[\begin{array}{c}{\cos 2t-\sin 2t}\\[4pt]{\cos 2t+\sin 2t}\\[4pt]{2\cos 2t}\end{array}\right]+c_{3}e^{4t}\left[\begin{array}{c}{\sin 2t+\cos 2t}\\[4pt]{\sin 2t-\cos 2t}\\[4pt]{2\sin 2t}\end{array}\right]\)
6. \({\bf y}=c_{1}\left[\begin{array}{c}{-1}\\[4pt]{-1}\\[4pt]{1}\end{array}\right]e^{-t}+c_{2}e^{-2t}\left[\begin{array}{c}{\cos 2t-\sin 2t}\\[4pt]{-\cos 2t-\sin 2t}\\[4pt]{2\cos 2t}\end{array}\right]+c_{3}e^{-2t}\left[\begin{array}{c}{\sin 2t+\cos 2t}\\[4pt]{-\sin 2t+\cos 2t}\\[4pt]{2\sin 2t}\end{array}\right]\)
7. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\[4pt]{1}\\[4pt]{1}\end{array}\right]e^{2t}+c_{2}e^{t}\left[\begin{array}{c}{-\sin t}\\[4pt]{\sin t}\\[4pt]{\cos t}\end{array}\right]+c_{3}e^{t}\left[\begin{array}{c}{\cos t}\\[4pt]{-\cos t}\\[4pt]{\sin t}\end{array}\right]\)
8. \({\bf y}=c_{1}\left[\begin{array}{c}{-1}\\[4pt]{1}\\[4pt]{1}\end{array}\right]e^{t}+c_{2}e^{-t}\left[\begin{array}{c}{-\sin 2t-\cos 2t}\\[4pt]{2\cos 2t}\\[4pt]{2\cos 2t}\end{array}\right]+c_{3}e^{-t}\left[\begin{array}{c}{\cos 2t-\sin 2t}\\[4pt]{2\sin 2t}\\[4pt]{2\sin 2t}\end{array}\right]\)
9. \({\bf y}=c_{1}e^{3t}\left[\begin{array}{c}{\cos 6t-3\sin 6t}\\[4pt]{5\cos 6t}\end{array}\right]+c_{2}e^{3t}\left[\begin{array}{c}{\sin 6t+3\cos 6t}\\[4pt]{5\sin 6t}\end{array}\right]\)
10. \({\bf y}=c_{1}e^{2t}\left[\begin{array}{c}{\cos t-3\sin t}\\[4pt]{2\cos t}\end{array}\right]+c_{2}e^{2t}\left[\begin{array}{c}{\sin t+3\cos t}\\[4pt]{2\sin t}\end{array}\right]\)
11. \({\bf y}=c_{1}e^{2t}\left[\begin{array}{c}{3\sin 3t-\cos 3t}\\[4pt]{5\cos 3t}\end{array}\right]+c_{2}e^{2t}\left[\begin{array}{c}{-3\cos 3t-\sin 3t}\\[4pt]{5\sin 3t}\end{array}\right]\)
12. \({\bf y}=c_{1}e^{2t}\left[\begin{array}{c}{\sin 4t-8\cos 4t}\\[4pt]{5\cos 4t}\end{array}\right]+c_{2}e^{2t}\left[\begin{array}{c}{-\cos 4t-8\sin 4t}\\[4pt]{5\sin 4t}\end{array}\right]\)
13. \({\bf y}=c_{1}\left[\begin{array}{c}{-1}\\[4pt]{1}\\[4pt]{1}\end{array}\right]e^{-2t}+c_{2}e^{t}\left[\begin{array}{c}{\sin t}\\[4pt]{-\cos t}\\[4pt]{\cos t}\end{array}\right]+c_{3}e^{t}\left[\begin{array}{c}{-\cos t}\\[4pt]{-\sin t}\\[4pt]{\sin t}\end{array}\right]\)
14. \({\bf y}=c_{1}\left[\begin{array}{c}{2}\\[4pt]{2}\\[4pt]{1}\end{array}\right]e^{-2t}+c_{2}e^{2t}\left[\begin{array}{c}{-\cos 3t-\sin 3t}\\[4pt]{-\sin 3t}\\[4pt]{\cos 3t}\end{array}\right]+c_{3}e^{2t}\left[\begin{array}{c}{-\sin 3t+\cos 3t}\\[4pt]{\cos 3t}\\[4pt]{\sin 3t}\end{array}\right]\)
15. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\[4pt]{2}\\[4pt]{1}\end{array}\right]e^{3t}+c_{2}e^{6t}\left[\begin{array}{c}{-\sin 3t}\\[4pt]{\sin 3t}\\[4pt]{\cos 3t}\end{array}\right]+c_{3}e^{6t}\left[\begin{array}{c}{\cos 3t}\\[4pt]{-\cos 3t}\\[4pt]{\sin 3t}\end{array}\right]\)
16. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\[4pt]{1}\\[4pt]{1}\end{array}\right]e^{t}+c_{2}e^{t}\left[\begin{array}{c}{2\cos t-2\sin t}\\[4pt]{\cos t-\sin t}\\[4pt]{2\cos t}\end{array}\right]+c_{3}e^{t}\left[\begin{array}{c}{2\sin t+2\cos t}\\[4pt]{\cos t+\sin t}\\[4pt]{2\sin t}\end{array}\right]\)
17. \({\bf y}=e^{t}\left[\begin{array}{c}{5\cos 3t+\sin 3t}\\[4pt]{2\cos 3t+3\sin 3t}\end{array}\right]\)
18. \({\bf y}=e^{4t}\left[\begin{array}{c}{5\cos 6t+5\sin 6t}\\[4pt]{\cos 6t-3\sin 6t}\end{array}\right]\)
19. \({\bf y}=e^{t}\left[\begin{array}{c}{17\cos 3t-\sin 3t}\\[4pt]{7\cos 3t+3\sin 3t}\end{array}\right]\)
20. \({\bf y}=e^{t/2}\left[\begin{array}{c}{\cos (t/2)+\sin (t/2)}\\[4pt]{-\cos (t/2)+2\sin (t/2)}\end{array}\right]\)
21. \({\bf y}=\left[\begin{array}{c}{1}\\[4pt]{-1}\\[4pt]{2}\end{array}\right]e^{t}+e^{4t}\left[\begin{array}{c}{3\cos t+\sin t}\\[4pt]{\cos t-3\sin t}\\[4pt]{4\cos t-2\sin t}\end{array}\right]\)
22. \({\bf y}=\left[\begin{array}{c}{4}\\[4pt]{4}\\[4pt]{2}\end{array}\right]e^{8t}+e^{2t}\left[\begin{array}{c}{4\cos 2t+8\sin 2t}\\[4pt]{-6\sin 2t+2\cos 2t}\\[4pt]{3\cos 2t+\sin 3t}\end{array}\right]\)
23. \({\bf y}=\left[\begin{array}{c}{0}\\[4pt]{3}\\[4pt]{3}\end{array}\right]e^{-4t}+e^{4t}\left[\begin{array}{c}{15\cos 6t+10\sin 6t}\\[4pt]{14\cos 6t-8\sin 6t}\\[4pt]{7\cos 6t-4\sin 6t}\end{array}\right]\)
24. \({\bf y}=\left[\begin{array}{c}{6}\\[4pt]{-3}\\[4pt]{3}\end{array}\right]e^{8t}+\left[\begin{array}{c}{10\cos 4t-4\sin 4t}\\[4pt]{17\cos 4t-\sin 4t}\\[4pt]{3\cos 4t-7\sin 4t}\end{array}\right]\)
29. \({\bf U}=\frac{1}{\sqrt{2}}\left[\begin{array}{c}{-1}\\[4pt]{1}\end{array}\right],\quad {\bf V}=\frac{1}{\sqrt{2}}\left[\begin{array}{c}{1}\\[4pt]{1}\end{array}\right]\)
30. \({\bf U}\approx \left[\begin{array}{c}{.5257}\\[4pt]{.8507}\end{array}\right],\quad {\bf V}\approx \left[\begin{array}{c}{-.8507}\\[4pt]{.5257}\end{array}\right]\)
31. \({\bf U}\approx \left[\begin{array}{c}{.8507}\\[4pt]{.5257}\end{array}\right],\quad {\bf V}\approx \left[\begin{array}{c}{-.5257}\\[4pt]{.8507}\end{array}\right]\)
32. \({\bf U}\approx \left[\begin{array}{c}{-.9732}\\[4pt]{.2298}\end{array}\right],\quad {\bf V}\approx \left[\begin{array}{c}{.2298}\\[4pt]{.9732}\end{array}\right]\)
33. \({\bf U}\approx \left[\begin{array}{c}{.5257}\\[4pt]{.8507}\end{array}\right],\quad {\bf V}\approx \left[\begin{array}{c}{-.8507}\\[4pt]{.5257}\end{array}\right]\)
34. \({\bf U}\approx \left[\begin{array}{c}{-.5257}\\[4pt]{.8507}\end{array}\right],\quad {\bf V}\approx \left[\begin{array}{c}{.8507}\\[4pt]{.5257}\end{array}\right]\)
35. \({\bf U}\approx \left[\begin{array}{c}{-.8817}\\[4pt]{.4719}\end{array}\right],\quad {\bf V}\approx \left[\begin{array}{c}{.4719}\\[4pt]{.8817}\end{array}\right]\)
36. \({\bf U}\approx \left[\begin{array}{c}{.8817}\\[4pt]{.4719}\end{array}\right],\quad {\bf V}\approx \left[\begin{array}{c}{-.4719}\\[4pt]{.8817}\end{array}\right]\)
37. \({\bf U}= \left[\begin{array}{c}{0}\\[4pt]{1}\end{array}\right],\quad {\bf V}= \left[\begin{array}{c}{-1}\\[4pt]{0}\end{array}\right]\)
38. \({\bf U}= \left[\begin{array}{c}{0}\\[4pt]{1}\end{array}\right],\quad {\bf V}= \left[\begin{array}{c}{1}\\[4pt]{0}\end{array}\right]\)
39. \({\bf U}=\frac{1}{\sqrt{2}}\left[\begin{array}{c}{1}\\[4pt]{1}\end{array}\right],\quad {\bf V}=\frac{1}{\sqrt{2}}\left[\begin{array}{c}{-1}\\[4pt]{1}\end{array}\right]\)
40. \({\bf U}\approx \left[\begin{array}{c}{.5257}\\[4pt]{.8507}\end{array}\right],\quad {\bf V}\approx \left[\begin{array}{c}{-.8507}\\[4pt]{.5257}\end{array}\right]\)