11.5: A.10.4 Section 10.4 Answers

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 \(L_{1}\).