40: Section 10.4 Answers

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

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

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

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

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

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

7. \({\bf y}=c_{1}\left[\begin{array}{c}{-2}\\{1}\end{array}\right]e^{-9t}+c_{2}\left(\left[\begin{array}{c}{-1}\\{0}\end{array}\right]e^{-9t}+\left[\begin{array}{c}{-2}\\{1}\end{array}\right]te^{-9t} \right)\)

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

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

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

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

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

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

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

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

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

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

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

19. \({\bf y}=\left[\begin{array}{c}{-1}\\{6}\end{array}\right]e^{4t}+\left[\begin{array}{c}{26}\\{13}\end{array}\right]te^{4t}\)

20. \({\bf y}=\left[\begin{array}{c}{6}\\{2}\end{array}\right]e^{-7t}- \left[\begin{array}{c}{8}\\{4}\end{array}\right]te^{-7t}\)

21. \({\bf y}=\left[\begin{array}{c}{5}\\{8}\end{array}\right]e^{3t}- \left[\begin{array}{c}{12}\\{16}\end{array}\right]te^{3t}\)

22. \({\bf y}=\left[\begin{array}{c}{2}\\{3}\end{array}\right]e^{-5t}- \left[\begin{array}{c}{8}\\{4}\end{array}\right]te^{-5t}\)

23. \({\bf y}=\left[\begin{array}{c}{3}\\{1}\end{array}\right]e^{5t}- \left[\begin{array}{c}{12}\\{6}\end{array}\right]te^{5t}\)

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

25. \({\bf y}=\left[\begin{array}{c}{4}\\{8}\\{-6}\end{array}\right]e^{t}+ \left[\begin{array}{c}{2}\\{-3}\\{-1}\end{array}\right]e^{-2t}+\left[\begin{array}{c}{-1}\\{1}\\{0}\end{array}\right]te^{-2t}\)

26. \({\bf y}=\left[\begin{array}{c}{3}\\{3}\\{6}\end{array}\right]e^{2t}- \left[\begin{array}{c}{9}\\{5}\\{6}\end{array}\right]+\left[\begin{array}{c}{2}\\{2}\\{0}\end{array}\right]t\)

27. \({\bf y}=-\left[\begin{array}{c}{2}\\{0}\\{2}\end{array}\right]e^{-3t}+ \left[\begin{array}{c}{-4}\\{9}\\{1}\end{array}\right]e^{t}-\left[\begin{array}{c}{0}\\{4}\\{4}\end{array}\right]te^{t}\)

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

29. \({\bf y}=-\left[\begin{array}{c}{1}\\{1}\\{0}\end{array}\right]e^{-4t}+ \left[\begin{array}{c}{-3}\\{2}\\{-3}\end{array}\right]e^{8t}+\left[\begin{array}{c}{8}\\{0}\\{-8}\end{array}\right]te^{8t}\)

30. \({\bf y}=\left[\begin{array}{c}{3}\\{6}\\{3}\end{array}\right]e^{4t}- \left[\begin{array}{c}{3}\\{4}\\{1}\end{array}\right]+\left[\begin{array}{c}{8}\\{4}\\{4}\end{array}\right]t\)

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

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

33. \({\bf y}=c_{1}\left[\begin{array}{c}{0}\\{1}\\{1}\end{array}\right]e^{6t} + c_{2}\left(\left[\begin{array}{c}{-1}\\{1}\\{0}\end{array}\right]\frac{e^{6t}}{4}+\left[\begin{array}{c}{0}\\{1}\\{1}\end{array}\right]te^{6t} \right)+c_{3}\left(\left[\begin{array}{c}{1}\\{1}\\{0}\end{array}\right]\frac{e^{6t}}{8}+\left[\begin{array}{c}{-1}\\{1}\\{0}\end{array}\right]\frac{te^{6t}}{4}+\left[\begin{array}{c}{0}\\{1}\\{1}\end{array}\right]\frac{t^{2}e^{6t}}{2} \right)\)

34. \({\bf y}=c_{1}\left[\begin{array}{c}{-1}\\{1}\\{1}\end{array}\right]e^{3t} + c_{2}\left(\left[\begin{array}{c}{1}\\{0}\\{0}\end{array}\right]\frac{e^{3t}}{2}+\left[\begin{array}{c}{-1}\\{1}\\{1}\end{array}\right]te^{3t} \right)+c_{3}\left(\left[\begin{array}{c}{1}\\{2}\\{0}\end{array}\right]\frac{e^{3t}}{36}+\left[\begin{array}{c}{1}\\{0}\\{0}\end{array}\right]\frac{te^{3t}}{2}+\left[\begin{array}{c}{-1}\\{1}\\{1}\end{array}\right]\frac{t^{2}e^{3t}}{2} \right)\)

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

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

37. \({\bf y}=c_{1}\left[\begin{array}{c}{-2}\\{1}\\{2}\end{array}\right]e^{-6t} + c_{2}\left(-\left[\begin{array}{c}{6}\\{1}\\{0}\end{array}\right]\frac{e^{-6t}}{6}+\left[\begin{array}{c}{-2}\\{1}\\{2}\end{array}\right]te^{-6t} \right)+c_{3}\left(-\left[\begin{array}{c}{12}\\{1}\\{0}\end{array}\right]\frac{e^{-6t}}{36}-\left[\begin{array}{c}{6}\\{1}\\{0}\end{array}\right]\frac{te^{-6t}}{6}+\left[\begin{array}{c}{-2}\\{1}\\{2}\end{array}\right]\frac{t^{2}e^{-6t}}{2} \right)\)

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

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

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

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

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