42: Section 10.6 Answers

1. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\{1}\end{array}\right]+c_{2}\left[\begin{array}{c}{3}\\{2}\end{array}\right]e^{t}-\left[\begin{array}{c}{11}\\{11}\end{array}\right]t-\left[\begin{array}{c}{15}\\{10}\end{array}\right]\)

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

3. \({\bf y}=c_{1}\left[\begin{array}{c}{10}\\{3}\end{array}\right]e^{3t/2}+c_{2}\left[\begin{array}{c}{2}\\{1}\end{array}\right]e^{t/2}+\left[\begin{array}{c}{-13/2}\\{-13/4}\end{array}\right]te^{t/2}+\left[\begin{array}{c}{-15/2}\\{-9/4}\end{array}\right]e^{t/2}\)

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

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

6. \({\bf y}=c_{1}\left[\begin{array}{c}{2}\\{1}\end{array}\right]e^t+c_{2}\left[\begin{array}{c}{1}\\{1}\end{array}\right]e^{2t}+\left[\begin{array}{c}{1/10}\\{-3/20}\end{array}\right]e^{-3t}-\left[\begin{array}{c}{3}\\{1}\end{array}\right]\)

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

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

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

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

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

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

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

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

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

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

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

18. \({\bf y}=c_{1}\left[\begin{array}{c}{\cos t-\sin t}\\{\cos t}\end{array}\right]+c_{2}\left[\begin{array}{c}{\cos t+\sin t}\\{\sin t}\end{array}\right]+\left[\begin{array}{c}{3\sin t\cos t-\cos^2 t-2\sin^2 t+(\sin t-\cos t)\ln|\sec t+\tan t|}\\{\sin^2 t-\cos^2 t-\cos t\ln|\sec t+\tan t|}\end{array}\right]\)

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

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

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

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

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