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Mathematics LibreTexts

10.6E: Constant Coefficient Homogeneous Systems III (Exercises)

  • Page ID
    18302
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    Q10.6.1

    In Exercises 10.6.1-10.6.16 find the general solution.

    1. \({\bf y}'=\left[\begin{array}{cc}{-1}&{2}\\{-5}&{5}\end{array}\right]{\bf y}\)

    2. \({\bf y}'=\left[\begin{array}{cc}{-11}&{4}\\{-26}&{9}\end{array}\right]{\bf y}\)

    3. \({\bf y}'=\left[\begin{array}{cc}{1}&{2}\\{-4}&{5}\end{array}\right]{\bf y}\)

    4. \({\bf y}'=\left[\begin{array}{cc}{5}&{-6}\\{3}&{-1}\end{array}\right]{\bf y}\)

    5. \({\bf y}'=\left[\begin{array}{ccc}{3}&{-3}&{1}\\{0}&{2}&{2}\\{5}&{1}&{1}\end{array}\right]{\bf y}\)

    6. \({\bf y}'=\left[\begin{array}{ccc}{-3}&{3}&{1}\\{1}&{-5}&{-3}\\{-3}&{7}&{3}\end{array}\right]{\bf y}\)

    7. \({\bf y}'=\left[\begin{array}{ccc}{2}&{1}&{-1}\\{0}&{1}&{1}\\{1}&{0}&{1}\end{array}\right]{\bf y}\)

    8. \({\bf y}'=\left[\begin{array}{ccc}{-3}&{1}&{-3}\\{4}&{-1}&{2}\\{4}&{-2}&{3}\end{array}\right]{\bf y}\)

    9. \({\bf y}'=\left[\begin{array}{cc}{5}&{-4}\\{10}&{1}\end{array}\right]{\bf y}\)

    10. \({\bf y}'=\frac{1}{3}\left[\begin{array}{cc}{7}&{-5}\\{2}&{5}\end{array}\right]{\bf y}\)

    11. \({\bf y}'=\left[\begin{array}{cc}{3}&{2}\\{-5}&{1}\end{array}\right]{\bf y}\)

    12. \({\bf y}'=\left[\begin{array}{cc}{34}&{52}\\{-20}&{-30}\end{array}\right]{\bf y}\)

    13. \({\bf y}'=\left[\begin{array}{ccc}{1}&{1}&{2}\\{1}&{0}&{-1}\\{-1}&{-2}&{-1}\end{array}\right]{\bf y}\)

    14. \({\bf y}'=\left[\begin{array}{ccc}{3}&{-4}&{-2}\\{-5}&{7}&{-8}\\{-10}&{13}&{-8}\end{array}\right]{\bf y}\)

    15. \({\bf y}'=\left[\begin{array}{ccc}{6}&{0}&{-3}\\{-3}&{3}&{3}\\{1}&{-2}&{6}\end{array}\right]{\bf y}\)

    16. \({\bf y}'=\left[\begin{array}{ccc}{1}&{2}&{-2}\\{0}&{2}&{-1}\\{1}&{0}&{0}\end{array}\right]{\bf y}\)

    Q10.6.2

    In Exercises 10.6.17-10.6.24 solve the initial value problem.

    17. \({\bf y}'=\left[\begin{array}{cc}{4}&{-6}\\{3}&{-2}\end{array}\right]{\bf y},\quad{\bf y}(0)=\left[\begin{array}{c}{5}\\{2}\end{array}\right]\)

    18. \({\bf y}'=\left[\begin{array}{cc}{7}&{15}\\{-3}&{1}\end{array}\right]{\bf y},\quad{\bf y}(0)=\left[\begin{array}{c}{5}\\{1}\end{array}\right]\)

    19. \({\bf y}'=\left[\begin{array}{cc}{7}&{-15}\\{3}&{-5}\end{array}\right]{\bf y},\quad{\bf y}(0)=\left[\begin{array}{c}{17}\\{7}\end{array}\right]\)

    20. \({\bf y}'=\frac{1}{6}\left[\begin{array}{cc}{4}&{-2}\\{5}&{2}\end{array}\right]{\bf y},\quad{\bf y}(0)=\left[\begin{array}{c}{1}\\{-1}\end{array}\right]\)

    21. \({\bf y}'=\left[\begin{array}{ccc}{5}&{2}&{-1}\\{-3}&{2}&{2}\\{1}&{3}&{2}\end{array}\right]{\bf y},\quad{\bf y}(0)=\left[\begin{array}{c}{4}\\{0}\\{6}\end{array}\right]\)

    22. \({\bf y}'=\left[\begin{array}{ccc}{4}&{4}&{0}\\{8}&{10}&{-20}\\{2}&{3}&{-2}\end{array}\right]{\bf y},\quad{\bf y}(0)=\left[\begin{array}{c}{8}\\{6}\\{5}\end{array}\right]\)

    23. \({\bf y}'=\left[\begin{array}{ccc}{1}&{15}&{-15}\\{-6}&{18}&{-22}\\{-3}&{11}&{-15}\end{array}\right]{\bf y},\quad{\bf y}(0)=\left[\begin{array}{c}{15}\\{17}\\{10}\end{array}\right]\)

    24. \({\bf y}'=\left[\begin{array}{ccc}{4}&{-4}&{4}\\{-10}&{3}&{15}\\{2}&{-3}&{1}\end{array}\right]{\bf y},\quad{\bf y}(0)=\left[\begin{array}{c}{16}\\{14}\\{6}\end{array}\right]\)

    Q10.6.3

    25. Suppose an \(n\times n\) matrix \(A\) with real entries has a complex eigenvalue \(\lambda=\alpha+i\beta\) (\(\beta\ne0\)) with associated eigenvector \({\bf x}={\bf u}+i{\bf v}\), where \({\bf u}\) and \({\bf v}\) have real components. Show that \({\bf u}\) and \({\bf v}\) are both nonzero.

    26. Verify that

    \[\bf y_1=e^{\alpha t}({\bf u}\cos\beta t-{\bf v}\sin\beta t) \quad \text{and}\quad \bf y_2=e^{\alpha t}({\bf u}\sin\beta t+{\bf v}\cos\beta t),\nonumber\]

    are the real and imaginary parts of

    \[e^{\alpha t}(\cos\beta t+i\sin\beta t)({\bf u}+i{\bf v}).\nonumber\]

    27. Show that if the vectors \({\bf u}\) and \({\bf v}\) are not both \({\bf 0}\) and \(\beta\ne0\) then the vector functions

    \[\bf y_1=e^{\alpha t}({\bf u}\cos\beta t-{\bf v}\sin\beta t)\quad \mbox{ and }\quad \bf y_2=e^{\alpha t}({\bf u}\sin\beta t+{\bf v}\cos\beta t)\nonumber\]

    are linearly independent on every interval.

    28. Suppose \({\bf u}=\left[\begin{array}{c}{u_{1}}\\{u_{2}}\end{array}\right]\) and \({\bf v}=\left[\begin{array}{c}{v_{1}}\\{v_{2}}\end{array}\right]\) are not orthogonal; that is, \(({\bf u},{\bf v})\ne0\).

    1. Show that the quadratic equation \[({\bf u},{\bf v})k^2+(\|{\bf v}\|^2-\|{\bf u}\|^2)k-({\bf u},{\bf v})=0\nonumber\] has a positive root \(k_1\) and a negative root \(k_2=-1/k_1\).
    2. Let \({\bf u}_1^{(1)}={\bf u}-k_1{\bf v}\), \({\bf v}_1^{(1)}={\bf v}+k_1{\bf u}\), \({\bf u}_1^{(2)}={\bf u}-k_2{\bf v}\), and \({\bf v}_1^{(2)}={\bf v}+k_2{\bf u}\), so that \(({\bf u}_1^{(1)},{\bf v}_1^{(1)}) =({\bf u}_1^{(2)},{\bf v}_1^{(2)})=0\), from the discussion given above. Show that \[{\bf u}_1^{(2)}={{\bf v}_1^{(1)}\over k_1} \quad \text{and} \quad {\bf v}_1^{(2)}=-{{\bf u}_1^{(1)}\over k_1}.\nonumber\]
    3. Let \({\bf U}_1\), \({\bf V}_1\), \({\bf U}_2\), and \({\bf V}_2\) be unit vectors in the directions of \({\bf u}_1^{(1)}\), \({\bf v}_1^{(1)}\), \({\bf u}_1^{(2)}\), and \({\bf v}_1^{(2)}\), respectively. Conclude from (a) that \({\bf U}_2={\bf V}_1\) and \({\bf V}_2=-{\bf U}_1\), and that therefore the counterclockwise angles from \({\bf U}_1\) to \({\bf V}_1\) and from \({\bf U}_2\) to \({\bf V}_2\) are both \(\pi/2\) or both \(-\pi/2\).

    Q10.6.4

    In Exercises 10.6.29-10.6.32 find vectors \({\bf U}\) and \({\bf V}\) parallel to the axes of symmetry of the trajectories, and plot some typical trajectories.

    29. \({\bf y}'=\left[\begin{array}{cc}{3}&{-5}\\{5}&{-3}\end{array}\right]{\bf y}\)

    30. \({\bf y}'=\left[\begin{array}{cc}{-15}&{10}\\{-25}&{15}\end{array}\right]{\bf y}\)

    31. \({\bf y}'=\left[\begin{array}{cc}{-4}&{8}\\{-4}&{4}\end{array}\right]{\bf y}\)

    32. \({\bf y}'=\left[\begin{array}{cc}{-3}&{-15}\\{3}&{3}\end{array}\right]{\bf y}\)

    Q10.6.5

    In Exercises 10.6.33-10.6.40 find vectors \({\bf U}\) and \({\bf V}\) parallel to the axes of symmetry of the shadow trajectories, and plot a typical trajectory.

    33. \({\bf y}'=\left[\begin{array}{cc}{-5}&{6}\\{-12}&{7}\end{array}\right]{\bf y}\)

    34. \({\bf y}'=\left[\begin{array}{cc}{5}&{-12}\\{6}&{-7}\end{array}\right]{\bf y}\)

    35. \({\bf y}'=\left[\begin{array}{cc}{4}&{-5}\\{9}&{-2}\end{array}\right]{\bf y}\)

    36. \({\bf y}'=\left[\begin{array}{cc}{-4}&{9}\\{-5}&{2}\end{array}\right]{\bf y}\)

    37. \({\bf y}'=\left[\begin{array}{cc}{-1}&{10}\\{-10}&{-1}\end{array}\right]{\bf y}\)

    38. \({\bf y}'=\left[\begin{array}{cc}{-1}&{-5}\\{20}&{-1}\end{array}\right]{\bf y}\)

    39. \({\bf y}'=\left[\begin{array}{cc}{-7}&{10}\\{-10}&{9}\end{array}\right]{\bf y}\)

    40. \({\bf y}'=\left[\begin{array}{cc}{-7}&{6}\\{-12}&{5}\end{array}\right]{\bf y}\)