41: Section 10.5 Answers

1. \({\bf y}=c_{1}e^{4t}\left[\begin{array}{c}{2\cos t-\sin t}\\{5\cos t}\end{array}\right]+c_{2}e^{4t}\left[\begin{array}{c}{2\sin t+\cos t}\\{5\sin t}\end{array}\right]\)

2. \({\bf y}=c_{1}\left[\begin{array}{c}{\sin t-\cos t}\\{2\cos t}\end{array}\right]+c_{2}\left[\begin{array}{c}{-\cos t-\sin t}\\{2\sin t}\end{array}\right]\)

3. \({\bf y}=c_{1}e^{4t}\left[\begin{array}{c}{\sin t-\cos t}\\{2\cos t}\end{array}\right]+c_{2}e^{4t}\left[\begin{array}{c}{-\sin t-\cos t}\\{2\sin t}\end{array}\right]\)

4. \({\bf y}=c_{1}e^{5t}\left[\begin{array}{c}{\cos 3t+3\sin 3t}\\{2\cos 3t}\end{array}\right]+c_{2}e^{5t}\left[\begin{array}{c}{\sin 3t-3\cos 3t}\\{2\cos 3t}\end{array}\right]\)

5. \({\bf y}=c_{1}\left[\begin{array}{c}{4\cos 3t-3\sin 3t}\\{5\cos 3t}\end{array}\right]+c_{2}\left[\begin{array}{c}{4\sin 3t+3\cos 3t}\\{5\sin 3t}\end{array}\right]\)

6. \({\bf y}=c_{1}e^{-t}\left[\begin{array}{c}{2\cos 2t-2\sin 2t}\\{\cos 2t}\end{array}\right]+c_{2}e^{-t}\left[\begin{array}{c}{2\cos 2t+2\sin 2t}\\{\sin 2t}\end{array}\right]\)

7. \({\bf y}=c_{1}e^{2t}\left[\begin{array}{c}{3\cos t+\sin t}\\{5\cos t}\end{array}\right]+c_{2}e^{2t}\left[\begin{array}{c}{3\sin t-\cos t}\\{5\sin t}\end{array}\right]\)

8. \({\bf y}=c_{1}e^{-t}\left[\begin{array}{c}{5\cos 2t+\sin 2t}\\{13\cos 2t}\end{array}\right]+c_{2}e^{-t}\left[\begin{array}{c}{5\sin 2t-\cos 2t}\\{13\sin 2t}\end{array}\right]\)

9. \({\bf y}=c_{1}e^{3t}\left[\begin{array}{c}{\cos 2t+\sin 2t}\\{2\cos 2t}\end{array}\right]+c_{2}e^{3t}\left[\begin{array}{c}{\sin 2t-\cos 2t}\\{2\sin 2t}\end{array}\right]\)

10. \({\bf y}=c_{1}e^{2t}\left[\begin{array}{c}{\cos 3t-\sin 3t}\\{\cos 3t}\end{array}\right]+c_{2}e^{2t}\left[\begin{array}{c}{\sin 3t+\cos 3t}\\{\sin 3t}\end{array}\right]\)

11. \({\bf y}=c_{1}e^{3t}\left[\begin{array}{c}{\cos 6t-3\sin 6t}\\{5\cos 6t}\end{array}\right]+c_{2}e^{3t}\left[\begin{array}{c}{\sin 6t+3\cos 6t}\\{5\sin 6t}\end{array}\right]\)

12. \({\bf y}=c_{1}e^{2t}\left[\begin{array}{c}{\cos t-3\sin t}\\{2\cos t}\end{array}\right]+c_{2}e^{2t}\left[\begin{array}{c}{\sin t+3\cos t}\\{2\sin t}\end{array}\right]\)

13. \({\bf y}=c_{1}e^{2t}\left[\begin{array}{c}{3\sin 3t-\cos 3t}\\{5\cos 3t}\end{array}\right]+c_{2}e^{2t}\left[\begin{array}{c}{-3\cos 3t-\sin 3t}\\{5\sin 3t}\end{array}\right]\)

14. \({\bf y}=c_{1}e^{2t}\left[\begin{array}{c}{\sin 4t-8\cos 4t}\\{5\cos 4t}\end{array}\right]+c_{2}e^{2t}\left[\begin{array}{c}{-\cos 4t-8\sin 4t}\\{5\sin 4t}\end{array}\right]\)

15. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\{0}\\{0}\end{array}\right]+c_{2}\left[\begin{array}{c}{\sin t}\\{-\sin t}\\{\cos t}\end{array}\right]+c_{3}\left[\begin{array}{c}{-\cos t}\\{\cos t}\\{\sin t}\end{array}\right]\)

16. \({\bf y}=c_{1}\left[\begin{array}{c}{0}\\{-2}\\{1}\end{array}\right]e^{-3t}+c_{2}e^{t}\left[\begin{array}{c}{-2\cos 2t+\sin 2t}\\{3\sin 2t}\\{2\cos 2t}\end{array}\right]+c_{3}e^{t}\left[\begin{array}{c}{-\cos 2t-2\sin 2t}\\{-3\cos 2t}\\{2\sin 2t}\end{array}\right]\)

17. \({\bf y}=c_{1}\left[\begin{array}{c}{0}\\{2}\\{1}\end{array}\right]e^{t}+c_{2}e^{t}\left[\begin{array}{c}{\cos t}\\{-\sin t}\\{-\sin t}\end{array}\right]+c_{3}e^{t}\left[\begin{array}{c}{\sin t}\\{\cos t}\\{\cos t}\end{array}\right]\)

18. \({\bf y}=c_{1}\left[\begin{array}{c}{0}\\{1}\\{0}\end{array}\right]e^{6t}+c_{2}e^{4t}\left[\begin{array}{c}{\sin 2t}\\{0}\\{2\cos 2t}\end{array}\right]+c_{3}e^{4t}\left[\begin{array}{c}{-\cos 2t}\\{0}\\{2\sin 2t}\end{array}\right]\)

19. \({\bf y}=c_{1}\left[\begin{array}{c}{28}\\{-5}\\{25}\end{array}\right]e^{2t}+c_{2}e^{-2t}\left[\begin{array}{c}{4\cos 3t-3\sin 3t}\\{-5\cos 3t}\\{0}\end{array}\right]+c_{3}e^{-2t}\left[\begin{array}{c}{4\sin 3t+3\cos 3t}\\{-5\sin 3t}\\{0}\end{array}\right]\)

20. \({\bf y}=c_{1}\left[\begin{array}{c}{0}\\{-1}\\{1}\end{array}\right]e^{-2t}+c_{2}\left[\begin{array}{c}{-2\cos 2t+2\sin 2t}\\{\cos 2t}\\{\cos 2t}\end{array}\right]+c_{3}\left[\begin{array}{c}{-2\cos 2t-2\sin 2t}\\{\sin 2t}\\{\sin 2t}\end{array}\right]\)

21. \({\bf y}=c_{1}\left[\begin{array}{c}{-1}\\{-1}\\{2}\end{array}\right]e^{-2t}+c_{2}e^{4t}\left[\begin{array}{c}{\cos 2t-\sin 2t}\\{\cos 2t+\sin 2t}\\{2\cos 2t}\end{array}\right]+c_{3}e^{4t}\left[\begin{array}{c}{\sin 2t+\cos 2t}\\{\sin 2t-\cos 2t}\\{2\sin 2t}\end{array}\right]\)

22. \({\bf y}=c_{1}\left[\begin{array}{c}{-1}\\{-1}\\{1}\end{array}\right]e^{-t}+c_{2}e^{-2t}\left[\begin{array}{c}{\cos 2t-\sin 2t}\\{-\cos 2t-\sin 2t}\\{2\cos 2t}\end{array}\right]+c_{3}e^{-2t}\left[\begin{array}{c}{\sin 2t+\cos 2t}\\{-\sin 2t+\cos 2t}\\{2\sin 2t}\end{array}\right]\)

23. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\{1}\\{1}\end{array}\right]e^{2t}+c_{2}e^{t}\left[\begin{array}{c}{-\sin t}\\{\sin t}\\{\cos t}\end{array}\right]+c_{3}e^{t}\left[\begin{array}{c}{\cos t}\\{-\cos t}\\{\sin t}\end{array}\right]\)

24. \({\bf y}=c_{1}\left[\begin{array}{c}{-1}\\{1}\\{1}\end{array}\right]e^{t}+c_{2}e^{-t}\left[\begin{array}{c}{-\sin 2t-\cos 2t}\\{2\cos 2t}\\{2\cos 2t}\end{array}\right]+c_{3}e^{-t}\left[\begin{array}{c}{\cos 2t-\sin 2t}\\{2\sin 2t}\\{2\sin 2t}\end{array}\right]\)

25. \({\bf y}=c_{1}\left[\begin{array}{c}{-1}\\{1}\\{1}\end{array}\right]e^{-2t}+c_{2}e^{t}\left[\begin{array}{c}{\sin t}\\{-\cos t}\\{\cos t}\end{array}\right]+c_{3}e^{t}\left[\begin{array}{c}{-\cos t}\\{-\sin t}\\{\sin t}\end{array}\right]\)

26. \({\bf y}=c_{1}\left[\begin{array}{c}{2}\\{2}\\{1}\end{array}\right]e^{-2t}+c_{2}e^{2t}\left[\begin{array}{c}{-\cos 3t-\sin 3t}\\{-\sin 3t}\\{\cos 3t}\end{array}\right]+c_{3}e^{2t}\left[\begin{array}{c}{-\sin 3t+\cos 3t}\\{\cos 3t}\\{\sin 3t}\end{array}\right]\)

27. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\{2}\\{1}\end{array}\right]e^{3t}+c_{2}e^{6t}\left[\begin{array}{c}{-\sin 3t}\\{\sin 3t}\\{\cos 3t}\end{array}\right]+c_{3}e^{6t}\left[\begin{array}{c}{\cos 3t}\\{-\cos 3t}\\{\sin 3t}\end{array}\right]\)

28. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\{1}\\{1}\end{array}\right]e^{t}+c_{2}e^{t}\left[\begin{array}{c}{2\cos t-2\sin t}\\{\cos t-\sin t}\\{2\cos t}\end{array}\right]+c_{3}e^{t}\left[\begin{array}{c}{2\sin t+2\cos t}\\{\cos t+\sin t}\\{2\sin t}\end{array}\right]\)

29. \({\bf y}=e^{5t}\left[\begin{array}{c}{-2\cos 2t+5\sin 2t}\\{-12\cos 2t+\sin 2t}\end{array}\right]\)

30. \({\bf y}=e^{t}\left[\begin{array}{c}{5\cos 3t+\sin 3t}\\{2\cos 3t+3\sin 3t}\end{array}\right]\)

31. \({\bf y}=e^{4t}\left[\begin{array}{c}{5\cos 6t+5\sin 6t}\\{\cos 6t-3\sin 6t}\end{array}\right]\)

32. \({\bf y}=e^{t}\left[\begin{array}{c}{17\cos 3t-\sin 3t}\\{7\cos 3t+3\sin 3t}\end{array}\right]\)

33. \({\bf y}=e^{t/2}\left[\begin{array}{c}{\cos (t/2)+\sin (t/2)}\\{-\cos (t/2)+2\sin (t/2)}\end{array}\right]\)

34. \({\bf y}=\left[\begin{array}{c}{-25}\\{7}\\{-6}\end{array}\right]e^{t}+\left[\begin{array}{c}{29\cos 5t+11\sin t}\\{-\cos 5t+6\sin 5t}\\{-\cos 5t+6\sin 5t}\end{array}\right]\)

35. \({\bf y}=\left[\begin{array}{c}{1}\\{-1}\\{2}\end{array}\right]e^{t}+e^{4t}\left[\begin{array}{c}{3\cos t+\sin t}\\{\cos t-3\sin t}\\{4\cos t-2\sin t}\end{array}\right]\)

36. \({\bf y}=\left[\begin{array}{c}{4}\\{4}\\{2}\end{array}\right]e^{8t}+e^{2t}\left[\begin{array}{c}{4\cos 2t+8\sin 2t}\\{-6\sin 2t+2\cos 2t}\\{3\cos 2t+\sin 3t}\end{array}\right]\)

37. \({\bf y}=\left[\begin{array}{c}{0}\\{3}\\{3}\end{array}\right]e^{-4t}+e^{4t}\left[\begin{array}{c}{15\cos 6t+10\sin 6t}\\{14\cos 6t-8\sin 6t}\\{7\cos 6t-4\sin 6t}\end{array}\right]\)

38. \({\bf y}=\left[\begin{array}{c}{6}\\{-3}\\{3}\end{array}\right]e^{8t}+\left[\begin{array}{c}{10\cos 4t-4\sin 4t}\\{17\cos 4t-\sin 4t}\\{3\cos 4t-7\sin 4t}\end{array}\right]\)

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