14.52: Section 10.6 Answers

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

2. \({\bf y}=c_{1}e^{-t}\left[\begin{array}{c}{5\cos 2t+\sin 2t}\\[4pt]{13\cos 2t}\end{array}\right]+c_{2}e^{-t}\left[\begin{array}{c}{5\sin 2t-\cos 2t}\\[4pt]{13\sin 2t}\end{array}\right]\)

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

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

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

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

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

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

9. \({\bf y}=c_{1}e^{3t}\left[\begin{array}{c}{\cos 6t-3\sin 6t}\\[4pt]{5\cos 6t}\end{array}\right]+c_{2}e^{3t}\left[\begin{array}{c}{\sin 6t+3\cos 6t}\\[4pt]{5\sin 6t}\end{array}\right]\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

29. \({\bf U}=\frac{1}{\sqrt{2}}\left[\begin{array}{c}{-1}\\[4pt]{1}\end{array}\right],\quad {\bf V}=\frac{1}{\sqrt{2}}\left[\begin{array}{c}{1}\\[4pt]{1}\end{array}\right]\)

30. \({\bf U}\approx \left[\begin{array}{c}{.5257}\\[4pt]{.8507}\end{array}\right],\quad {\bf V}\approx \left[\begin{array}{c}{-.8507}\\[4pt]{.5257}\end{array}\right]\)

31. \({\bf U}\approx \left[\begin{array}{c}{.8507}\\[4pt]{.5257}\end{array}\right],\quad {\bf V}\approx \left[\begin{array}{c}{-.5257}\\[4pt]{.8507}\end{array}\right]\)

32. \({\bf U}\approx \left[\begin{array}{c}{-.9732}\\[4pt]{.2298}\end{array}\right],\quad {\bf V}\approx \left[\begin{array}{c}{.2298}\\[4pt]{.9732}\end{array}\right]\)

33. \({\bf U}\approx \left[\begin{array}{c}{.5257}\\[4pt]{.8507}\end{array}\right],\quad {\bf V}\approx \left[\begin{array}{c}{-.8507}\\[4pt]{.5257}\end{array}\right]\)

34. \({\bf U}\approx \left[\begin{array}{c}{-.5257}\\[4pt]{.8507}\end{array}\right],\quad {\bf V}\approx \left[\begin{array}{c}{.8507}\\[4pt]{.5257}\end{array}\right]\)

35. \({\bf U}\approx \left[\begin{array}{c}{-.8817}\\[4pt]{.4719}\end{array}\right],\quad {\bf V}\approx \left[\begin{array}{c}{.4719}\\[4pt]{.8817}\end{array}\right]\)

36. \({\bf U}\approx \left[\begin{array}{c}{.8817}\\[4pt]{.4719}\end{array}\right],\quad {\bf V}\approx \left[\begin{array}{c}{-.4719}\\[4pt]{.8817}\end{array}\right]\)

37. \({\bf U}= \left[\begin{array}{c}{0}\\[4pt]{1}\end{array}\right],\quad {\bf V}= \left[\begin{array}{c}{-1}\\[4pt]{0}\end{array}\right]\)

38. \({\bf U}= \left[\begin{array}{c}{0}\\[4pt]{1}\end{array}\right],\quad {\bf V}= \left[\begin{array}{c}{1}\\[4pt]{0}\end{array}\right]\)

39. \({\bf U}=\frac{1}{\sqrt{2}}\left[\begin{array}{c}{1}\\[4pt]{1}\end{array}\right],\quad {\bf V}=\frac{1}{\sqrt{2}}\left[\begin{array}{c}{-1}\\[4pt]{1}\end{array}\right]\)

40. \({\bf U}\approx \left[\begin{array}{c}{.5257}\\[4pt]{.8507}\end{array}\right],\quad {\bf V}\approx \left[\begin{array}{c}{-.8507}\\[4pt]{.5257}\end{array}\right]\)

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