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Mathematics LibreTexts

Section 10.6 Answers

  • Page ID
    30070
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    1. \({\bf y}=c_{1}e^{2t}\left[\begin{array}{c}{3\cos t+\sin t}\\{5\cos t}\end{array}\right]+c_{2}e^{2t}\left[\begin{array}{c}{3\sin t-\cos t}\\{5\sin t}\end{array}\right]\)

    2.  \({\bf y}=c_{1}e^{-t}\left[\begin{array}{c}{5\cos 2t+\sin 2t}\\{13\cos 2t}\end{array}\right]+c_{2}e^{-t}\left[\begin{array}{c}{5\sin 2t-\cos 2t}\\{13\sin 2t}\end{array}\right]\)

    3.  \({\bf y}=c_{1}e^{3t}\left[\begin{array}{c}{\cos 2t+\sin 2t}\\{2\cos 2t}\end{array}\right]+c_{2}e^{3t}\left[\begin{array}{c}{\sin 2t-\cos 2t}\\{2\sin 2t}\end{array}\right]\)

    4.  \({\bf y}=c_{1}e^{2t}\left[\begin{array}{c}{\cos 3t-\sin 3t}\\{\cos 3t}\end{array}\right]+c_{2}e^{2t}\left[\begin{array}{c}{\sin 3t+\cos 3t}\\{\sin 3t}\end{array}\right]\)

    5. \({\bf y}=c_{1}\left[\begin{array}{c}{-1}\\{-1}\\{2}\end{array}\right]e^{-2t}+c_{2}e^{4t}\left[\begin{array}{c}{\cos 2t-\sin 2t}\\{\cos 2t+\sin 2t}\\{2\cos 2t}\end{array}\right]+c_{3}e^{4t}\left[\begin{array}{c}{\sin 2t+\cos 2t}\\{\sin 2t-\cos 2t}\\{2\sin 2t}\end{array}\right]\)

    6.  \({\bf y}=c_{1}\left[\begin{array}{c}{-1}\\{-1}\\{1}\end{array}\right]e^{-t}+c_{2}e^{-2t}\left[\begin{array}{c}{\cos 2t-\sin 2t}\\{-\cos 2t-\sin 2t}\\{2\cos 2t}\end{array}\right]+c_{3}e^{-2t}\left[\begin{array}{c}{\sin 2t+\cos 2t}\\{-\sin 2t+\cos 2t}\\{2\sin 2t}\end{array}\right]\)

    7.  \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\{1}\\{1}\end{array}\right]e^{2t}+c_{2}e^{t}\left[\begin{array}{c}{-\sin t}\\{\sin t}\\{\cos t}\end{array}\right]+c_{3}e^{t}\left[\begin{array}{c}{\cos t}\\{-\cos t}\\{\sin t}\end{array}\right]\)

    8.  \({\bf y}=c_{1}\left[\begin{array}{c}{-1}\\{1}\\{1}\end{array}\right]e^{t}+c_{2}e^{-t}\left[\begin{array}{c}{-\sin 2t-\cos 2t}\\{2\cos 2t}\\{2\cos 2t}\end{array}\right]+c_{3}e^{-t}\left[\begin{array}{c}{\cos 2t-\sin 2t}\\{2\sin 2t}\\{2\sin 2t}\end{array}\right]\)

    9. \({\bf y}=c_{1}e^{3t}\left[\begin{array}{c}{\cos 6t-3\sin 6t}\\{5\cos 6t}\end{array}\right]+c_{2}e^{3t}\left[\begin{array}{c}{\sin 6t+3\cos 6t}\\{5\sin 6t}\end{array}\right]\)

    10.  \({\bf y}=c_{1}e^{2t}\left[\begin{array}{c}{\cos t-3\sin t}\\{2\cos t}\end{array}\right]+c_{2}e^{2t}\left[\begin{array}{c}{\sin t+3\cos t}\\{2\sin t}\end{array}\right]\)

    11.  \({\bf y}=c_{1}e^{2t}\left[\begin{array}{c}{3\sin 3t-\cos 3t}\\{5\cos 3t}\end{array}\right]+c_{2}e^{2t}\left[\begin{array}{c}{-3\cos 3t-\sin 3t}\\{5\sin 3t}\end{array}\right]\)

    12.  \({\bf y}=c_{1}e^{2t}\left[\begin{array}{c}{\sin 4t-8\cos 4t}\\{5\cos 4t}\end{array}\right]+c_{2}e^{2t}\left[\begin{array}{c}{-\cos 4t-8\sin 4t}\\{5\sin 4t}\end{array}\right]\)

    13. \({\bf y}=c_{1}\left[\begin{array}{c}{-1}\\{1}\\{1}\end{array}\right]e^{-2t}+c_{2}e^{t}\left[\begin{array}{c}{\sin t}\\{-\cos t}\\{\cos t}\end{array}\right]+c_{3}e^{t}\left[\begin{array}{c}{-\cos t}\\{-\sin t}\\{\sin t}\end{array}\right]\)

    14. \({\bf y}=c_{1}\left[\begin{array}{c}{2}\\{2}\\{1}\end{array}\right]e^{-2t}+c_{2}e^{2t}\left[\begin{array}{c}{-\cos 3t-\sin 3t}\\{-\sin 3t}\\{\cos 3t}\end{array}\right]+c_{3}e^{2t}\left[\begin{array}{c}{-\sin 3t+\cos 3t}\\{\cos 3t}\\{\sin 3t}\end{array}\right]\)

    15. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\{2}\\{1}\end{array}\right]e^{3t}+c_{2}e^{6t}\left[\begin{array}{c}{-\sin 3t}\\{\sin 3t}\\{\cos 3t}\end{array}\right]+c_{3}e^{6t}\left[\begin{array}{c}{\cos 3t}\\{-\cos 3t}\\{\sin 3t}\end{array}\right]\)

    16. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\{1}\\{1}\end{array}\right]e^{t}+c_{2}e^{t}\left[\begin{array}{c}{2\cos t-2\sin t}\\{\cos t-\sin t}\\{2\cos t}\end{array}\right]+c_{3}e^{t}\left[\begin{array}{c}{2\sin t+2\cos t}\\{\cos t+\sin t}\\{2\sin t}\end{array}\right]\)

    17. \({\bf y}=e^{t}\left[\begin{array}{c}{5\cos 3t+\sin 3t}\\{2\cos 3t+3\sin 3t}\end{array}\right]\)

    18. \({\bf y}=e^{4t}\left[\begin{array}{c}{5\cos 6t+5\sin 6t}\\{\cos 6t-3\sin 6t}\end{array}\right]\)

    19. \({\bf y}=e^{t}\left[\begin{array}{c}{17\cos 3t-\sin 3t}\\{7\cos 3t+3\sin 3t}\end{array}\right]\)

    20. \({\bf y}=e^{t/2}\left[\begin{array}{c}{\cos (t/2)+\sin (t/2)}\\{-\cos (t/2)+2\sin (t/2)}\end{array}\right]\)

    21. \({\bf y}=\left[\begin{array}{c}{1}\\{-1}\\{2}\end{array}\right]e^{t}+e^{4t}\left[\begin{array}{c}{3\cos t+\sin t}\\{\cos t-3\sin t}\\{4\cos t-2\sin t}\end{array}\right]\)

    22.  \({\bf y}=\left[\begin{array}{c}{4}\\{4}\\{2}\end{array}\right]e^{8t}+e^{2t}\left[\begin{array}{c}{4\cos 2t+8\sin 2t}\\{-6\sin 2t+2\cos 2t}\\{3\cos 2t+\sin 3t}\end{array}\right]\)

    23.  \({\bf y}=\left[\begin{array}{c}{0}\\{3}\\{3}\end{array}\right]e^{-4t}+e^{4t}\left[\begin{array}{c}{15\cos 6t+10\sin 6t}\\{14\cos 6t-8\sin 6t}\\{7\cos 6t-4\sin 6t}\end{array}\right]\)

    24.  \({\bf y}=\left[\begin{array}{c}{6}\\{-3}\\{3}\end{array}\right]e^{8t}+\left[\begin{array}{c}{10\cos 4t-4\sin 4t}\\{17\cos 4t-\sin 4t}\\{3\cos 4t-7\sin 4t}\end{array}\right]\)

    29. \({\bf U}=\frac{1}{\sqrt{2}}\left[\begin{array}{c}{-1}\\{1}\end{array}\right],\quad {\bf V}=\frac{1}{\sqrt{2}}\left[\begin{array}{c}{1}\\{1}\end{array}\right]\)

    30. \({\bf U}\approx \left[\begin{array}{c}{.5257}\\{.8507}\end{array}\right],\quad {\bf V}\approx \left[\begin{array}{c}{-.8507}\\{.5257}\end{array}\right]\)

    31. \({\bf U}\approx \left[\begin{array}{c}{.8507}\\{.5257}\end{array}\right],\quad {\bf V}\approx \left[\begin{array}{c}{-.5257}\\{.8507}\end{array}\right]\)

    32.  \({\bf U}\approx \left[\begin{array}{c}{-.9732}\\{.2298}\end{array}\right],\quad {\bf V}\approx \left[\begin{array}{c}{.2298}\\{.9732}\end{array}\right]\)

    33.  \({\bf U}\approx \left[\begin{array}{c}{.5257}\\{.8507}\end{array}\right],\quad {\bf V}\approx \left[\begin{array}{c}{-.8507}\\{.5257}\end{array}\right]\)

    34.  \({\bf U}\approx \left[\begin{array}{c}{-.5257}\\{.8507}\end{array}\right],\quad {\bf V}\approx \left[\begin{array}{c}{.8507}\\{.5257}\end{array}\right]\)

    35.  \({\bf U}\approx \left[\begin{array}{c}{-.8817}\\{.4719}\end{array}\right],\quad {\bf V}\approx \left[\begin{array}{c}{.4719}\\{.8817}\end{array}\right]\)

    36.  \({\bf U}\approx \left[\begin{array}{c}{.8817}\\{.4719}\end{array}\right],\quad {\bf V}\approx \left[\begin{array}{c}{-.4719}\\{.8817}\end{array}\right]\)

    37.  \({\bf U}= \left[\begin{array}{c}{0}\\{1}\end{array}\right],\quad {\bf V}= \left[\begin{array}{c}{-1}\\{0}\end{array}\right]\)

    38. \({\bf U}= \left[\begin{array}{c}{0}\\{1}\end{array}\right],\quad {\bf V}= \left[\begin{array}{c}{1}\\{0}\end{array}\right]\)

    39. \({\bf U}=\frac{1}{\sqrt{2}}\left[\begin{array}{c}{1}\\{1}\end{array}\right],\quad {\bf V}=\frac{1}{\sqrt{2}}\left[\begin{array}{c}{-1}\\{1}\end{array}\right]\)

    40. \({\bf U}\approx \left[\begin{array}{c}{.5257}\\{.8507}\end{array}\right],\quad {\bf V}\approx \left[\begin{array}{c}{-.8507}\\{.5257}\end{array}\right]\)