39: Section 10.3 Answers

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

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

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

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

5. \({\bf y}=c_{1}\left[\begin{array}{c}{5}\\{2}\end{array}\right]e^{8t}+c_{2}\left[\begin{array}{c}{1}\\{4}\end{array}\right]e^{-10t}\)

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

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

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

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

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

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

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

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

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

15. \({\bf y}=c_{1}\left[\begin{array}{c}{4}\\{0}\\{-5}\end{array}\right]e^{2t}+c_{2}\left[\begin{array}{c}{-7}\\{3}\\{5}\end{array}\right]e^{5t}+c_{3}\left[\begin{array}{c}{-7}\\{5}\\{5}\end{array}\right]e^{7t}\)

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

17. \({\bf y}=c_{1}\left[\begin{array}{c}{2}\\{1}\\{2}\end{array}\right]e^{8t}+c_{2}\left[\begin{array}{c}{0}\\{-2}\\{1}\end{array}\right]e^{-t}+c_{3}\left[\begin{array}{c}{1}\\{-2}\\{0}\end{array}\right]e^{-t}\)

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

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

20. \({\bf y}=c_{1}\left[\begin{array}{c}{4}\\{0}\\{-1}\end{array}\right]e^{-t}+c_{2}\left[\begin{array}{c}{-12}\\{6}\\{5}\end{array}\right]e^{-t/2}+c_{3}\left[\begin{array}{c}{4}\\{2}\\{-1}\end{array}\right]e^{-3t/2}\)

21. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\{1}\\{0}\end{array}\right]e^{3t}+c_{2}\left[\begin{array}{c}{1}\\{-1}\\{0}\end{array}\right]e^{-5t}+c_{3}\left[\begin{array}{c}{2}\\{-2}\\{11}\end{array}\right]e^{6t}\)

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

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

24. \({\bf y}=c_{1}\left[\begin{array}{c}{-2}\\{-4}\\{3}\end{array}\right]e^{t}+c_{2}\left[\begin{array}{c}{-1}\\{1}\\{0}\end{array}\right]e^{-2t}+c_{3}\left[\begin{array}{c}{-7}\\{-5}\\{4}\end{array}\right]e^{2t}\)

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

26. \({\bf y}=c_{1}\left[\begin{array}{c}{11}\\{7}\\{1}\end{array}\right]e^{3t}+c_{2}\left[\begin{array}{c}{1}\\{2}\\{1}\end{array}\right]e^{-2t}+c_{3}\left[\begin{array}{c}{1}\\{1}\\{1}\end{array}\right]e^{-t}\)

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

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

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

30. \({\bf y}=\left[\begin{array}{c}{0}\\{2}\end{array}\right]e^{-t/2}+\left[\begin{array}{c}{3}\\{3}\end{array}\right]e^{t/2}\)

31. \({\bf y}=-\left[\begin{array}{c}{2}\\{6}\end{array}\right]e^{5t}+\left[\begin{array}{c}{4}\\{2}\end{array}\right]e^{-5t}\)

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

33. \({\bf y}=\left[\begin{array}{c}{7}\\{7}\end{array}\right]e^{9t}-\left[\begin{array}{c}{2}\\{4}\end{array}\right]e^{-3t}\)

34. \({\bf y}=\left[\begin{array}{c}{3}\\{9}\end{array}\right]e^{5t}-\left[\begin{array}{c}{4}\\{2}\end{array}\right]e^{-5t}\)

35. \({\bf y}=\left[\begin{array}{c}{-5}\\{3}\\{-2}\end{array}\right]e^{2t}+\left[\begin{array}{c}{5}\\{0}\\{5\over 2}\end{array}\right]e^{3t}+\left[\begin{array}{c}{1}\\{0}\\-{1\over 2}\end{array}\right]e^{-t}\)

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

37. \({\bf y}=\left[\begin{array}{c}{5}\\{5}\\{0}\end{array}\right]e^{t/2}+\left[\begin{array}{c}{0}\\{0}\\{1}\end{array}\right]e^{t/2}+\left[\begin{array}{c}{-1}\\{2}\\{0}\end{array}\right]e^{-t/2}\)

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

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

40. \({\bf y}=-\left[\begin{array}{c}{1}\\{2}\\{1}\end{array}\right]e^{t}+\left[\begin{array}{c}{4}\\{2}\\{4}\end{array}\right]e^{-t}+\left[\begin{array}{c}{1}\\{1}\\{0}\end{array}\right]e^{2t}\)

41. \({\bf y}=\left[\begin{array}{c}{-2}\\{-2}\\{2}\end{array}\right]e^{2t}-\left[\begin{array}{c}{0}\\{3}\\{0}\end{array}\right]e^{-2t}+\left[\begin{array}{c}{4}\\{12}\\{4}\end{array}\right]e^{4t}\)

42. \({\bf y}=\left[\begin{array}{c}{-1}\\{-1}\\{1}\end{array}\right]e^{-6t}+\left[\begin{array}{c}{2}\\{-2}\\{2}\end{array}\right]e^{2t}+\left[\begin{array}{c}{7}\\{-7}\\{-7}\end{array}\right]e^{4t}\)

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

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

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