8.6E: Constant Coefficient Homogeneous Systems III (Exercises)

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Q10.6.1

In Exercises 10.6.1-10.6.16 find the general solution.

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

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

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

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

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

6. \({\bf y}'=\left[\begin{array}{ccc}{-3}&{3}&{1}\\[4pt]{1}&{-5}&{-3}\\[4pt]{-3}&{7}&{3}\end{array}\right]{\bf y}\)

7. \({\bf y}'=\left[\begin{array}{ccc}{2}&{1}&{-1}\\[4pt]{0}&{1}&{1}\\[4pt]{1}&{0}&{1}\end{array}\right]{\bf y}\)

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

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

10. \({\bf y}'=\frac{1}{3}\left[\begin{array}{cc}{7}&{-5}\\[4pt]{2}&{5}\end{array}\right]{\bf y}\)

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

12. \({\bf y}'=\left[\begin{array}{cc}{34}&{52}\\[4pt]{-20}&{-30}\end{array}\right]{\bf y}\)

13. \({\bf y}'=\left[\begin{array}{ccc}{1}&{1}&{2}\\[4pt]{1}&{0}&{-1}\\[4pt]{-1}&{-2}&{-1}\end{array}\right]{\bf y}\)

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

15. \({\bf y}'=\left[\begin{array}{ccc}{6}&{0}&{-3}\\[4pt]{-3}&{3}&{3}\\[4pt]{1}&{-2}&{6}\end{array}\right]{\bf y}\)

16. \({\bf y}'=\left[\begin{array}{ccc}{1}&{2}&{-2}\\[4pt]{0}&{2}&{-1}\\[4pt]{1}&{0}&{0}\end{array}\right]{\bf y}\)

Q10.6.2

In Exercises 10.6.17-10.6.24 solve the initial value problem.

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

18. \({\bf y}'=\left[\begin{array}{cc}{7}&{15}\\[4pt]{-3}&{1}\end{array}\right]{\bf y},\quad{\bf y}(0)=\left[\begin{array}{c}{5}\\[4pt]{1}\end{array}\right]\)

19. \({\bf y}'=\left[\begin{array}{cc}{7}&{-15}\\[4pt]{3}&{-5}\end{array}\right]{\bf y},\quad{\bf y}(0)=\left[\begin{array}{c}{17}\\[4pt]{7}\end{array}\right]\)

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

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

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

23. \({\bf y}'=\left[\begin{array}{ccc}{1}&{15}&{-15}\\[4pt]{-6}&{18}&{-22}\\[4pt]{-3}&{11}&{-15}\end{array}\right]{\bf y},\quad{\bf y}(0)=\left[\begin{array}{c}{15}\\[4pt]{17}\\[4pt]{10}\end{array}\right]\)

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

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 \({\bf u}=\left[\begin{array}{c}{u_{1}}\\[4pt]{u_{2}}\end{array}\right]\) and \({\bf v}=\left[\begin{array}{c}{v_{1}}\\[4pt]{v_{2}}\end{array}\right]\) 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. \({\bf y}'=\left[\begin{array}{cc}{3}&{-5}\\[4pt]{5}&{-3}\end{array}\right]{\bf y}\)

30. \({\bf y}'=\left[\begin{array}{cc}{-15}&{10}\\[4pt]{-25}&{15}\end{array}\right]{\bf y}\)

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

32. \({\bf y}'=\left[\begin{array}{cc}{-3}&{-15}\\[4pt]{3}&{3}\end{array}\right]{\bf y}\)

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. \({\bf y}'=\left[\begin{array}{cc}{-5}&{6}\\[4pt]{-12}&{7}\end{array}\right]{\bf y}\)

34. \({\bf y}'=\left[\begin{array}{cc}{5}&{-12}\\[4pt]{6}&{-7}\end{array}\right]{\bf y}\)

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

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

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

38. \({\bf y}'=\left[\begin{array}{cc}{-1}&{-5}\\[4pt]{20}&{-1}\end{array}\right]{\bf y}\)

39. \({\bf y}'=\left[\begin{array}{cc}{-7}&{10}\\[4pt]{-10}&{9}\end{array}\right]{\bf y}\)

40. \({\bf y}'=\left[\begin{array}{cc}{-7}&{6}\\[4pt]{-12}&{5}\end{array}\right]{\bf y}\)