A.10.4 Section 10.4 Answers
- Page ID
- 43737
1. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\{1}\end{array}\right]e^{3t}+c_{2}\left[\begin{array}{c}{1}\\{-1}\end{array}\right]e^{-t}\)
2. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\{1}\end{array}\right]e^{-t/2}+c_{2}\left[\begin{array}{c}{-1}\\{1}\end{array}\right]e^{-2t}\)
3. \({\bf y}=c_{1}\left[\begin{array}{c}{-3}\\{1}\end{array}\right]e^{-t}+c_{2}\left[\begin{array}{c}{-1}\\{2}\end{array}\right]e^{-2t}\)
4. \({\bf y}=c_{1}\left[\begin{array}{c}{2}\\{1}\end{array}\right]e^{-3t}+c_{2}\left[\begin{array}{c}{-2}\\{1}\end{array}\right]e^{t}\)
5. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\{1}\end{array}\right]e^{-2t}+c_{2}\left[\begin{array}{c}{-4}\\{1}\end{array}\right]e^{3t}\)
6. \({\bf y}=c_{1}\left[\begin{array}{c}{3}\\{2}\end{array}\right]e^{2t}+c_{2}\left[\begin{array}{c}{1}\\{1}\end{array}\right]e^{t}\)
7. \({\bf y}=c_{1}\left[\begin{array}{c}{-3}\\{1}\end{array}\right]e^{-5t}+c_{2}\left[\begin{array}{c}{-1}\\{1}\end{array}\right]e^{-3t}\)
8. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\{2}\\{1}\end{array}\right]e^{-3t}+c_{2}\left[\begin{array}{c}{-1}\\{-4}\\{1}\end{array}\right]e^{-t}+c_{3}\left[\begin{array}{c}{-1}\\{-1}\\{1}\end{array}\right]e^{2t}\)
9. \({\bf y}=c_{1}\left[\begin{array}{c}{2}\\{1}\\{2}\end{array}\right]e^{-16t}+c_{2}\left[\begin{array}{c}{-1}\\{2}\\{0}\end{array}\right]e^{2t}+c_{3}\left[\begin{array}{c}{-1}\\{0}\\{1}\end{array}\right]e^{2t}\)
10. \({\bf y}=c_{1}\left[\begin{array}{c}{-2}\\{-4}\\{3}\end{array}\right]e^{t}+c_{2}\left[\begin{array}{c}{-1}\\{1}\\{0}\end{array}\right]e^{-2t}+c_{3}\left[\begin{array}{c}{-7}\\{-5}\\{4}\end{array}\right]e^{2t}\)
11. \({\bf y}=c_{1}\left[\begin{array}{c}{-1}\\{-1}\\{1}\end{array}\right]e^{-2t}+c_{2}\left[\begin{array}{c}{-1}\\{-2}\\{1}\end{array}\right]e^{-3t}+c_{3}\left[\begin{array}{c}{-2}\\{-6}\\{3}\end{array}\right]e^{-5t}\)
12. \({\bf y}=c_{1}\left[\begin{array}{c}{11}\\{7}\\{1}\end{array}\right]e^{3t}+c_{2}\left[\begin{array}{c}{1}\\{2}\\{1}\end{array}\right]e^{-2t}+c_{3}\left[\begin{array}{c}{1}\\{1}\\{1}\end{array}\right]e^{-t}\)
13. \({\bf y}=c_{1}\left[\begin{array}{c}{4}\\{-1}\\{1}\end{array}\right]e^{-4t}+c_{2}\left[\begin{array}{c}{-1}\\{-1}\\{1}\end{array}\right]e^{6t}+c_{3}\left[\begin{array}{c}{-1}\\{0}\\{1}\end{array}\right]e^{4t}\)
14. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\{1}\\{5}\end{array}\right]e^{-5t}+c_{2}\left[\begin{array}{c}{-1}\\{0}\\{1}\end{array}\right]e^{5t}+c_{3}\left[\begin{array}{c}{1}\\{1}\\{0}\end{array}\right]e^{5t}\)
15. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\{-1}\\{2}\end{array}\right]+c_{2}\left[\begin{array}{c}{-1}\\{0}\\{3}\end{array}\right]e^{6t}+c_{3}\left[\begin{array}{c}{1}\\{3}\\{0}\end{array}\right]e^{6t}\)
16. \({\bf y}=-\left[\begin{array}{c}{2}\\{6}\end{array}\right]e^{5t}+\left[\begin{array}{c}{4}\\{2}\end{array}\right]e^{-5t}\)
17. \({\bf y}=\left[\begin{array}{c}{2}\\{-4}\end{array}\right]e^{t/2}+\left[\begin{array}{c}{-2}\\{1}\end{array}\right]e^{t}\)
18. \({\bf y}=\left[\begin{array}{c}{7}\\{7}\end{array}\right]e^{9t}-\left[\begin{array}{c}{2}\\{4}\end{array}\right]e^{-3t}\)
19. \({\bf y}=\left[\begin{array}{c}{3}\\{9}\end{array}\right]e^{5t}-\left[\begin{array}{c}{4}\\{2}\end{array}\right]e^{-5t}\)
20. \({\bf y}=\left[\begin{array}{c}{5}\\{5}\\{0}\end{array}\right]e^{t/2}+\left[\begin{array}{c}{0}\\{0}\\{1}\end{array}\right]e^{t/2}+\left[\begin{array}{c}{-1}\\{2}\\{0}\end{array}\right]e^{-t/2}\)
21. \({\bf y}=\left[\begin{array}{c}{3}\\{3}\\{3}\end{array}\right]e^{t}+\left[\begin{array}{c}{-2}\\{-2}\\{2}\end{array}\right]e^{-t}\)
22. \({\bf y}=\left[\begin{array}{c}{2}\\{-2}\\{2}\end{array}\right]e^{t}-\left[\begin{array}{c}{3}\\{0}\\{3}\end{array}\right]e^{-2t}+\left[\begin{array}{c}{1}\\{1}\\{0}\end{array}\right]e^{3t}\)
23. \({\bf y}=-\left[\begin{array}{c}{1}\\{2}\\{1}\end{array}\right]e^{t}+\left[\begin{array}{c}{4}\\{2}\\{4}\end{array}\right]e^{-t}+\left[\begin{array}{c}{1}\\{1}\\{0}\end{array}\right]e^{2t}\)
24. \({\bf y}=\left[\begin{array}{c}{-2}\\{-2}\\{2}\end{array}\right]e^{2t}-\left[\begin{array}{c}{0}\\{3}\\{0}\end{array}\right]e^{-2t}+\left[\begin{array}{c}{4}\\{12}\\{4}\end{array}\right]e^{4t}\)
25. \({\bf y}=\left[\begin{array}{c}{-1}\\{-1}\\{1}\end{array}\right]e^{-6t}+\left[\begin{array}{c}{2}\\{-2}\\{2}\end{array}\right]e^{2t}+\left[\begin{array}{c}{7}\\{-7}\\{-7}\end{array}\right]e^{4t}\)
26. \({\bf y}=\left[\begin{array}{c}{1}\\{4}\\{4}\end{array}\right]e^{-t}+\left[\begin{array}{c}{6}\\{6}\\{-2}\end{array}\right]e^{2t}\)
27. \({\bf y}=\left[\begin{array}{c}{4}\\{-2}\\{2}\end{array}\right]+\left[\begin{array}{c}{3}\\{-9}\\{6}\end{array}\right]e^{4t}+\left[\begin{array}{c}{-1}\\{1}\\{-1}\end{array}\right]e^{2t}\)
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 \(\left[\begin{array}{c}{1}\\{-1}\end{array}\right]\), 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 \(\left[\begin{array}{c}{-1}\\{2}\end{array}\right]\), 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 \(\left[\begin{array}{c}{1}\\{-1}\end{array}\right]\),\(\left[\begin{array}{c}{-1}\\{1}\end{array}\right]\), 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 \(\left[\begin{array}{c}{1}\\{-1}\end{array}\right]\), \(L_{1}\).
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 \(\left[\begin{array}{c}{-4}\\{1}\end{array}\right]\), 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 \(\left[\begin{array}{c}{1}\\{-1}\end{array}\right]\), traversed away from \(L_{1}\).