A.10.5: Section 10.5 Answers

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

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

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

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

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

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

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

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

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

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

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

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

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

14. \({\bf y}=\left[\begin{array}{c}{5}\\[4pt]{8}\end{array}\right]e^{3t}- \left[\begin{array}{c}{12}\\[4pt]{16}\end{array}\right]te^{3t}\)

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

16. \({\bf y}=\left[\begin{array}{c}{3}\\[4pt]{1}\end{array}\right]e^{5t}- \left[\begin{array}{c}{12}\\[4pt]{6}\end{array}\right]te^{5t}\)

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

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

19. \({\bf y}=\left[\begin{array}{c}{3}\\[4pt]{3}\\[4pt]{6}\end{array}\right]e^{2t}- \left[\begin{array}{c}{9}\\[4pt]{5}\\[4pt]{6}\end{array}\right]+\left[\begin{array}{c}{2}\\[4pt]{2}\\[4pt]{0}\end{array}\right]t\)

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

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

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

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

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

25. \({\bf y}=c_{1}\left[\begin{array}{c}{-1}\\[4pt]{1}\\[4pt]{1}\end{array}\right]e^{3t} + c_{2}\left(\left[\begin{array}{c}{1}\\[4pt]{0}\\[4pt]{0}\end{array}\right]\frac{e^{3t}}{2}+\left[\begin{array}{c}{-1}\\[4pt]{1}\\[4pt]{1}\end{array}\right]te^{3t} \right)+c_{3}\left(\left[\begin{array}{c}{1}\\[4pt]{2}\\[4pt]{0}\end{array}\right]\frac{e^{3t}}{36}+\left[\begin{array}{c}{1}\\[4pt]{0}\\[4pt]{0}\end{array}\right]\frac{te^{3t}}{2}+\left[\begin{array}{c}{-1}\\[4pt]{1}\\[4pt]{1}\end{array}\right]\frac{t^{2}e^{3t}}{2} \right)\)

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

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

28. \({\bf y}=c_{1}\left[\begin{array}{c}{-2}\\[4pt]{1}\\[4pt]{2}\end{array}\right]e^{-6t} + c_{2}\left(-\left[\begin{array}{c}{6}\\[4pt]{1}\\[4pt]{0}\end{array}\right]\frac{e^{-6t}}{6}+\left[\begin{array}{c}{-2}\\[4pt]{1}\\[4pt]{2}\end{array}\right]te^{-6t} \right)+c_{3}\left(-\left[\begin{array}{c}{12}\\[4pt]{1}\\[4pt]{0}\end{array}\right]\frac{e^{-6t}}{36}-\left[\begin{array}{c}{6}\\[4pt]{1}\\[4pt]{0}\end{array}\right]\frac{te^{-6t}}{6}+\left[\begin{array}{c}{-2}\\[4pt]{1}\\[4pt]{2}\end{array}\right]\frac{t^{2}e^{-6t}}{2} \right)\)

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

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

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

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

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