14.6: Sección 10.4 Respuestas
- Page ID
- 115355
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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. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\{1}\end{array}\right]e^{3t}+c_{2}\left[\begin{array}{c}{1}\\{-1}\end{array}\right]e^{-t}\)
2. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\{1}\end{array}\right]e^{-t/2}+c_{2}\left[\begin{array}{c}{-1}\\{1}\end{array}\right]e^{-2t}\)
3. \({\bf y}=c_{1}\left[\begin{array}{c}{-3}\\{1}\end{array}\right]e^{-t}+c_{2}\left[\begin{array}{c}{-1}\\{2}\end{array}\right]e^{-2t}\)
4. \({\bf y}=c_{1}\left[\begin{array}{c}{2}\\{1}\end{array}\right]e^{-3t}+c_{2}\left[\begin{array}{c}{-2}\\{1}\end{array}\right]e^{t}\)
5. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\{1}\end{array}\right]e^{-2t}+c_{2}\left[\begin{array}{c}{-4}\\{1}\end{array}\right]e^{3t}\)
6. \({\bf y}=c_{1}\left[\begin{array}{c}{3}\\{2}\end{array}\right]e^{2t}+c_{2}\left[\begin{array}{c}{1}\\{1}\end{array}\right]e^{t}\)
7. \({\bf y}=c_{1}\left[\begin{array}{c}{-3}\\{1}\end{array}\right]e^{-5t}+c_{2}\left[\begin{array}{c}{-1}\\{1}\end{array}\right]e^{-3t}\)
8. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\{2}\\{1}\end{array}\right]e^{-3t}+c_{2}\left[\begin{array}{c}{-1}\\{-4}\\{1}\end{array}\right]e^{-t}+c_{3}\left[\begin{array}{c}{-1}\\{-1}\\{1}\end{array}\right]e^{2t}\)
9. \({\bf y}=c_{1}\left[\begin{array}{c}{2}\\{1}\\{2}\end{array}\right]e^{-16t}+c_{2}\left[\begin{array}{c}{-1}\\{2}\\{0}\end{array}\right]e^{2t}+c_{3}\left[\begin{array}{c}{-1}\\{0}\\{1}\end{array}\right]e^{2t}\)
10. \({\bf y}=c_{1}\left[\begin{array}{c}{-2}\\{-4}\\{3}\end{array}\right]e^{t}+c_{2}\left[\begin{array}{c}{-1}\\{1}\\{0}\end{array}\right]e^{-2t}+c_{3}\left[\begin{array}{c}{-7}\\{-5}\\{4}\end{array}\right]e^{2t}\)
11. \({\bf y}=c_{1}\left[\begin{array}{c}{-1}\\{-1}\\{1}\end{array}\right]e^{-2t}+c_{2}\left[\begin{array}{c}{-1}\\{-2}\\{1}\end{array}\right]e^{-3t}+c_{3}\left[\begin{array}{c}{-2}\\{-6}\\{3}\end{array}\right]e^{-5t}\)
12. \({\bf y}=c_{1}\left[\begin{array}{c}{11}\\{7}\\{1}\end{array}\right]e^{3t}+c_{2}\left[\begin{array}{c}{1}\\{2}\\{1}\end{array}\right]e^{-2t}+c_{3}\left[\begin{array}{c}{1}\\{1}\\{1}\end{array}\right]e^{-t}\)
13. \({\bf y}=c_{1}\left[\begin{array}{c}{4}\\{-1}\\{1}\end{array}\right]e^{-4t}+c_{2}\left[\begin{array}{c}{-1}\\{-1}\\{1}\end{array}\right]e^{6t}+c_{3}\left[\begin{array}{c}{-1}\\{0}\\{1}\end{array}\right]e^{4t}\)
14. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\{1}\\{5}\end{array}\right]e^{-5t}+c_{2}\left[\begin{array}{c}{-1}\\{0}\\{1}\end{array}\right]e^{5t}+c_{3}\left[\begin{array}{c}{1}\\{1}\\{0}\end{array}\right]e^{5t}\)
15. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\{-1}\\{2}\end{array}\right]+c_{2}\left[\begin{array}{c}{-1}\\{0}\\{3}\end{array}\right]e^{6t}+c_{3}\left[\begin{array}{c}{1}\\{3}\\{0}\end{array}\right]e^{6t}\)
16. \({\bf y}=-\left[\begin{array}{c}{2}\\{6}\end{array}\right]e^{5t}+\left[\begin{array}{c}{4}\\{2}\end{array}\right]e^{-5t}\)
17. \({\bf y}=\left[\begin{array}{c}{2}\\{-4}\end{array}\right]e^{t/2}+\left[\begin{array}{c}{-2}\\{1}\end{array}\right]e^{t}\)
18. \({\bf y}=\left[\begin{array}{c}{7}\\{7}\end{array}\right]e^{9t}-\left[\begin{array}{c}{2}\\{4}\end{array}\right]e^{-3t}\)
19. \({\bf y}=\left[\begin{array}{c}{3}\\{9}\end{array}\right]e^{5t}-\left[\begin{array}{c}{4}\\{2}\end{array}\right]e^{-5t}\)
20. \({\bf y}=\left[\begin{array}{c}{5}\\{5}\\{0}\end{array}\right]e^{t/2}+\left[\begin{array}{c}{0}\\{0}\\{1}\end{array}\right]e^{t/2}+\left[\begin{array}{c}{-1}\\{2}\\{0}\end{array}\right]e^{-t/2}\)
21. \({\bf y}=\left[\begin{array}{c}{3}\\{3}\\{3}\end{array}\right]e^{t}+\left[\begin{array}{c}{-2}\\{-2}\\{2}\end{array}\right]e^{-t}\)
22. \({\bf y}=\left[\begin{array}{c}{2}\\{-2}\\{2}\end{array}\right]e^{t}-\left[\begin{array}{c}{3}\\{0}\\{3}\end{array}\right]e^{-2t}+\left[\begin{array}{c}{1}\\{1}\\{0}\end{array}\right]e^{3t}\)
23. \({\bf y}=-\left[\begin{array}{c}{1}\\{2}\\{1}\end{array}\right]e^{t}+\left[\begin{array}{c}{4}\\{2}\\{4}\end{array}\right]e^{-t}+\left[\begin{array}{c}{1}\\{1}\\{0}\end{array}\right]e^{2t}\)
24. \({\bf y}=\left[\begin{array}{c}{-2}\\{-2}\\{2}\end{array}\right]e^{2t}-\left[\begin{array}{c}{0}\\{3}\\{0}\end{array}\right]e^{-2t}+\left[\begin{array}{c}{4}\\{12}\\{4}\end{array}\right]e^{4t}\)
25. \({\bf y}=\left[\begin{array}{c}{-1}\\{-1}\\{1}\end{array}\right]e^{-6t}+\left[\begin{array}{c}{2}\\{-2}\\{2}\end{array}\right]e^{2t}+\left[\begin{array}{c}{7}\\{-7}\\{-7}\end{array}\right]e^{4t}\)
26. \({\bf y}=\left[\begin{array}{c}{1}\\{4}\\{4}\end{array}\right]e^{-t}+\left[\begin{array}{c}{6}\\{6}\\{-2}\end{array}\right]e^{2t}\)
27. \({\bf y}=\left[\begin{array}{c}{4}\\{-2}\\{2}\end{array}\right]+\left[\begin{array}{c}{3}\\{-9}\\{6}\end{array}\right]e^{4t}+\left[\begin{array}{c}{-1}\\{1}\\{-1}\end{array}\right]e^{2t}\)
29. Medias líneas de\(L_{1} : y_{2} = y_{1}\) y\(L_{2} : y_{2} = −y_{1}\) son trayectorias otras trayectorias son asintóticamente tangentes a\(L_{1}\) as\(t → −∞\) y asintóticamente tangentes a\(L_{2}\) as\(t → ∞\).
30. Medias líneas de\(L_{1} : y_{2} = −2y_{1}\) y\(L_{2} : y_{2} = −y_{1}/3\) son trayectorias otras trayectorias son asintóticamente paralelas a\(L_{1}\) as\(t → −∞\) y asintóticamente tangentes a\(L_{2}\) as\(t → ∞\).
31. Las medias líneas de\(L_{1} : y_{2} = y_{1}/3\) y\(L_{2} : y_{2} = −y_{1}\) son trayectorias otras trayectorias son asintóticamente tangentes a\(L_{1}\) as\(t → −∞\) y asintóticamente paralelas a\(L_{2}\) as\(t → ∞\).
32. Medias líneas de\(L_{1} : y_{2} = y_{1}/2\) y\(L_{2} : y_{2} = −y_{1}\) son trayectorias otras trayectorias son asintóticamente tangentes a\(L_{1}\) as\(t → −∞\) y asintóticamente tangentes a\(L_{2}\) as\(t → ∞\).
33. Las medias líneas de\(L_{1} : y_{2} = −y_{1}/4\) y\(L_{2} : y_{2} = −y_{1}\) son trayectorias otras trayectorias son asintóticamente tangentes a\(L_{1}\) as\(t → −∞\) y asintóticamente paralelas a\(L_{2}\) as\(t → ∞\).
34. Medias líneas de\(L_{1} : y_{2} = −y_{1}\) y\(L_{2} : y_{2} = 3y_{1}\) son trayectorias otras trayectorias son asintóticamente paralelas a\(L_{1}\) as\(t → −∞\) y asintóticamente tangentes a\(L_{2}\) as\(t → ∞\).
36. Los puntos\(L_{2} : y_{2} = y_{1}\) son trayectorias de soluciones constantes. Las trayectorias de soluciones no constantes son medias líneas a cada lado de\(L_{1}\), paralelas a\(\left[\begin{array}{c}{1}\\{-1}\end{array}\right]\), atravesadas hacia L1.
37. Los puntos\(L_{1} : y_{2} = −y_{1}/3\) son trayectorias de soluciones constantes. Las trayectorias de las soluciones no constantes son medias líneas a cada lado de\(L_{1}\), paralelas a\(\left[\begin{array}{c}{-1}\\{2}\end{array}\right]\), atravesadas lejos de\(L_{1}\).
38. Los puntos\(L_{1} : y_{2} = y_{1}/3\) son trayectorias de soluciones constantes. Las trayectorias de las soluciones no constantes son medias líneas a cada lado de\(L_{1}\)\(\left[\begin{array}{c}{1}\\{-1}\end{array}\right]\), paralelas a\(\left[\begin{array}{c}{-1}\\{1}\end{array}\right]\), atravesadas lejos de\(L_{1}\).
39. Los puntos\(L_{1} : y_{2} = y_{1}/2\) son trayectorias de soluciones constantes. Las trayectorias de soluciones no constantes son medias líneas a cada lado de\(L_{1}\), paralelas a\(\left[\begin{array}{c}{1}\\{-1}\end{array}\right]\),\(L_{1}\).
40. Los puntos\(L_{2} : y_{2} = −y_{1}\) son trayectorias de soluciones constantes. Las trayectorias de soluciones no constantes son medias líneas a cada lado de\(L_{2}\), paralelas a\(\left[\begin{array}{c}{-4}\\{1}\end{array}\right]\), atravesadas hacia\(L_{1}\).
41. Los puntos\(L_{1} : y_{2} = 3y_{1}\) son trayectorias de soluciones constantes. Las trayectorias de las soluciones no constantes son medias líneas a cada lado de\(L_{1}\), paralelas a\(\left[\begin{array}{c}{1}\\{-1}\end{array}\right]\), atravesadas lejos de\(L_{1}\).