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11.5: A.10.4 Section 10.4 Answers

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    121462
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    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 \(L_{1}\).


    This page titled 11.5: A.10.4 Section 10.4 Answers is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.

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