A.10.4 Section 10.4 Answers
- Page ID
- 43737
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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. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\[4pt]{1}\end{array}\right]e^{3t}+c_{2}\left[\begin{array}{c}{1}\\[4pt]{-1}\end{array}\right]e^{-t}\)
2. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\[4pt]{1}\end{array}\right]e^{-t/2}+c_{2}\left[\begin{array}{c}{-1}\\[4pt]{1}\end{array}\right]e^{-2t}\)
3. \({\bf y}=c_{1}\left[\begin{array}{c}{-3}\\[4pt]{1}\end{array}\right]e^{-t}+c_{2}\left[\begin{array}{c}{-1}\\[4pt]{2}\end{array}\right]e^{-2t}\)
4. \({\bf y}=c_{1}\left[\begin{array}{c}{2}\\[4pt]{1}\end{array}\right]e^{-3t}+c_{2}\left[\begin{array}{c}{-2}\\[4pt]{1}\end{array}\right]e^{t}\)
5. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\[4pt]{1}\end{array}\right]e^{-2t}+c_{2}\left[\begin{array}{c}{-4}\\[4pt]{1}\end{array}\right]e^{3t}\)
6. \({\bf y}=c_{1}\left[\begin{array}{c}{3}\\[4pt]{2}\end{array}\right]e^{2t}+c_{2}\left[\begin{array}{c}{1}\\[4pt]{1}\end{array}\right]e^{t}\)
7. \({\bf y}=c_{1}\left[\begin{array}{c}{-3}\\[4pt]{1}\end{array}\right]e^{-5t}+c_{2}\left[\begin{array}{c}{-1}\\[4pt]{1}\end{array}\right]e^{-3t}\)
8. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\[4pt]{2}\\[4pt]{1}\end{array}\right]e^{-3t}+c_{2}\left[\begin{array}{c}{-1}\\[4pt]{-4}\\[4pt]{1}\end{array}\right]e^{-t}+c_{3}\left[\begin{array}{c}{-1}\\[4pt]{-1}\\[4pt]{1}\end{array}\right]e^{2t}\)
9. \({\bf y}=c_{1}\left[\begin{array}{c}{2}\\[4pt]{1}\\[4pt]{2}\end{array}\right]e^{-16t}+c_{2}\left[\begin{array}{c}{-1}\\[4pt]{2}\\[4pt]{0}\end{array}\right]e^{2t}+c_{3}\left[\begin{array}{c}{-1}\\[4pt]{0}\\[4pt]{1}\end{array}\right]e^{2t}\)
10. \({\bf y}=c_{1}\left[\begin{array}{c}{-2}\\[4pt]{-4}\\[4pt]{3}\end{array}\right]e^{t}+c_{2}\left[\begin{array}{c}{-1}\\[4pt]{1}\\[4pt]{0}\end{array}\right]e^{-2t}+c_{3}\left[\begin{array}{c}{-7}\\[4pt]{-5}\\[4pt]{4}\end{array}\right]e^{2t}\)
11. \({\bf y}=c_{1}\left[\begin{array}{c}{-1}\\[4pt]{-1}\\[4pt]{1}\end{array}\right]e^{-2t}+c_{2}\left[\begin{array}{c}{-1}\\[4pt]{-2}\\[4pt]{1}\end{array}\right]e^{-3t}+c_{3}\left[\begin{array}{c}{-2}\\[4pt]{-6}\\[4pt]{3}\end{array}\right]e^{-5t}\)
12. \({\bf y}=c_{1}\left[\begin{array}{c}{11}\\[4pt]{7}\\[4pt]{1}\end{array}\right]e^{3t}+c_{2}\left[\begin{array}{c}{1}\\[4pt]{2}\\[4pt]{1}\end{array}\right]e^{-2t}+c_{3}\left[\begin{array}{c}{1}\\[4pt]{1}\\[4pt]{1}\end{array}\right]e^{-t}\)
13. \({\bf y}=c_{1}\left[\begin{array}{c}{4}\\[4pt]{-1}\\[4pt]{1}\end{array}\right]e^{-4t}+c_{2}\left[\begin{array}{c}{-1}\\[4pt]{-1}\\[4pt]{1}\end{array}\right]e^{6t}+c_{3}\left[\begin{array}{c}{-1}\\[4pt]{0}\\[4pt]{1}\end{array}\right]e^{4t}\)
14. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\[4pt]{1}\\[4pt]{5}\end{array}\right]e^{-5t}+c_{2}\left[\begin{array}{c}{-1}\\[4pt]{0}\\[4pt]{1}\end{array}\right]e^{5t}+c_{3}\left[\begin{array}{c}{1}\\[4pt]{1}\\[4pt]{0}\end{array}\right]e^{5t}\)
15. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\[4pt]{-1}\\[4pt]{2}\end{array}\right]+c_{2}\left[\begin{array}{c}{-1}\\[4pt]{0}\\[4pt]{3}\end{array}\right]e^{6t}+c_{3}\left[\begin{array}{c}{1}\\[4pt]{3}\\[4pt]{0}\end{array}\right]e^{6t}\)
16. \({\bf y}=-\left[\begin{array}{c}{2}\\[4pt]{6}\end{array}\right]e^{5t}+\left[\begin{array}{c}{4}\\[4pt]{2}\end{array}\right]e^{-5t}\)
17. \({\bf y}=\left[\begin{array}{c}{2}\\[4pt]{-4}\end{array}\right]e^{t/2}+\left[\begin{array}{c}{-2}\\[4pt]{1}\end{array}\right]e^{t}\)
18. \({\bf y}=\left[\begin{array}{c}{7}\\[4pt]{7}\end{array}\right]e^{9t}-\left[\begin{array}{c}{2}\\[4pt]{4}\end{array}\right]e^{-3t}\)
19. \({\bf y}=\left[\begin{array}{c}{3}\\[4pt]{9}\end{array}\right]e^{5t}-\left[\begin{array}{c}{4}\\[4pt]{2}\end{array}\right]e^{-5t}\)
20. \({\bf y}=\left[\begin{array}{c}{5}\\[4pt]{5}\\[4pt]{0}\end{array}\right]e^{t/2}+\left[\begin{array}{c}{0}\\[4pt]{0}\\[4pt]{1}\end{array}\right]e^{t/2}+\left[\begin{array}{c}{-1}\\[4pt]{2}\\[4pt]{0}\end{array}\right]e^{-t/2}\)
21. \({\bf y}=\left[\begin{array}{c}{3}\\[4pt]{3}\\[4pt]{3}\end{array}\right]e^{t}+\left[\begin{array}{c}{-2}\\[4pt]{-2}\\[4pt]{2}\end{array}\right]e^{-t}\)
22. \({\bf y}=\left[\begin{array}{c}{2}\\[4pt]{-2}\\[4pt]{2}\end{array}\right]e^{t}-\left[\begin{array}{c}{3}\\[4pt]{0}\\[4pt]{3}\end{array}\right]e^{-2t}+\left[\begin{array}{c}{1}\\[4pt]{1}\\[4pt]{0}\end{array}\right]e^{3t}\)
23. \({\bf y}=-\left[\begin{array}{c}{1}\\[4pt]{2}\\[4pt]{1}\end{array}\right]e^{t}+\left[\begin{array}{c}{4}\\[4pt]{2}\\[4pt]{4}\end{array}\right]e^{-t}+\left[\begin{array}{c}{1}\\[4pt]{1}\\[4pt]{0}\end{array}\right]e^{2t}\)
24. \({\bf y}=\left[\begin{array}{c}{-2}\\[4pt]{-2}\\[4pt]{2}\end{array}\right]e^{2t}-\left[\begin{array}{c}{0}\\[4pt]{3}\\[4pt]{0}\end{array}\right]e^{-2t}+\left[\begin{array}{c}{4}\\[4pt]{12}\\[4pt]{4}\end{array}\right]e^{4t}\)
25. \({\bf y}=\left[\begin{array}{c}{-1}\\[4pt]{-1}\\[4pt]{1}\end{array}\right]e^{-6t}+\left[\begin{array}{c}{2}\\[4pt]{-2}\\[4pt]{2}\end{array}\right]e^{2t}+\left[\begin{array}{c}{7}\\[4pt]{-7}\\[4pt]{-7}\end{array}\right]e^{4t}\)
26. \({\bf y}=\left[\begin{array}{c}{1}\\[4pt]{4}\\[4pt]{4}\end{array}\right]e^{-t}+\left[\begin{array}{c}{6}\\[4pt]{6}\\[4pt]{-2}\end{array}\right]e^{2t}\)
27. \({\bf y}=\left[\begin{array}{c}{4}\\[4pt]{-2}\\[4pt]{2}\end{array}\right]+\left[\begin{array}{c}{3}\\[4pt]{-9}\\[4pt]{6}\end{array}\right]e^{4t}+\left[\begin{array}{c}{-1}\\[4pt]{1}\\[4pt]{-1}\end{array}\right]e^{2t}\)
29. Half lines of \(L_{1} : y_{2} = y_{1}\) and \(L_{2} : y_{2} = −y_{1}\) are trajectories other trajectories are asymptotically tangent to \(L_{1}\) as \(t → −∞\) and asymptotically tangent to \(L_{2}\) as \(t → ∞\).
30. Half lines of \(L_{1} : y_{2} = −2y_{1}\) and \(L_{2} : y_{2} = −y_{1}/3\) are trajectories other trajectories are asymptotically parallel to \(L_{1}\) as \(t → −∞\) and asymptotically tangent to \(L_{2}\) as \(t → ∞\).
31. Half lines of \(L_{1} : y_{2} = y_{1}/3\) and \(L_{2} : y_{2} = −y_{1}\) are trajectories other trajectories are asymptotically tangent to \(L_{1}\) as \(t → −∞\) and asymptotically parallel to \(L_{2}\) as \(t → ∞\).
32. Half lines of \(L_{1} : y_{2} = y_{1}/2\) and \(L_{2} : y_{2} = −y_{1}\) are trajectories other trajectories are asymptotically tangent to \(L_{1}\) as \(t → −∞\) and asymptotically tangent to \(L_{2}\) as \(t → ∞\).
33. Half lines of \(L_{1} : y_{2} = −y_{1}/4\) and \(L_{2} : y_{2} = −y_{1}\) are trajectories other trajectories are asymptotically tangent to \(L_{1}\) as \(t → −∞\) and asymptotically parallel to \(L_{2}\) as \(t → ∞\).
34. Half lines of \(L_{1} : y_{2} = −y_{1}\) and \(L_{2} : y_{2} = 3y_{1}\) are trajectories other trajectories are asymptotically parallel to \(L_{1}\) as \(t → −∞\) and asymptotically tangent to \(L_{2}\) as \(t → ∞\).
36. Points on \(L_{2} : y_{2} = y_{1}\) are trajectories of constant solutions. The trajectories of nonconstant solutions are half-lines on either side of \(L_{1}\), parallel to \(\left[\begin{array}{c}{1}\\[4pt]{-1}\end{array}\right]\), traversed toward L1.
37. Points on \(L_{1} : y_{2} = −y_{1}/3\) are trajectories of constant solutions. The trajectories of nonconstant solutions are half-lines on either side of \(L_{1}\), parallel to \(\left[\begin{array}{c}{-1}\\[4pt]{2}\end{array}\right]\), traversed away from \(L_{1}\).
38. Points on \(L_{1} : y_{2} = y_{1}/3\) are trajectories of constant solutions. The trajectories of nonconstant solutions are half-lines on either side of \(L_{1}\), parallel to \(\left[\begin{array}{c}{1}\\[4pt]{-1}\end{array}\right]\),\(\left[\begin{array}{c}{-1}\\[4pt]{1}\end{array}\right]\), traversed away from \(L_{1}\).
39. Points on \(L_{1} : y_{2} = y_{1}/2\) are trajectories of constant solutions. The trajectories of nonconstant solutions are half-lines on either side of \(L_{1}\), parallel to \(\left[\begin{array}{c}{1}\\[4pt]{-1}\end{array}\right]\), \(L_{1}\).
40. Points on \(L_{2} : y_{2} = −y_{1}\) are trajectories of constant solutions. The trajectories of nonconstant solutions are half-lines on either side of \(L_{2}\), parallel to \(\left[\begin{array}{c}{-4}\\[4pt]{1}\end{array}\right]\), traversed toward \(L_{1}\).
41. Points on \(L_{1} : y_{2} = 3y_{1}\) are trajectories of constant solutions. The trajectories of nonconstant solutions are half-lines on either side of \(L_{1}\), parallel to \(\left[\begin{array}{c}{1}\\[4pt]{-1}\end{array}\right]\), traversed away from \(L_{1}\).