14.53: Section 10.7 Answers

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1. \(\left[\begin{array}{c}{5e^{4t}+e^{-3t}(2+8t)}\\[4pt]{-e^{4t}-e^{-3t}(1-4t)}\end{array}\right]\)

2. \(\left[\begin{array}{c}{13e^{3t}+3e^{-3t}}\\[4pt]{-e^{3t}-11e^{-3t}}\end{array}\right]\)

3. \(\frac{1}{9}\left[\begin{array}{c}{7-6t}\\[4pt]{-11+3t}\end{array}\right]\)

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

5. \(\left[\begin{array}{c}{e^{-5t}(3+6t)+e^{-3t}(3-2t)}\\[4pt]{-e^{-5t}(3+2t)-e^{-3t}(1-2t)}\end{array}\right]\)

6. \(\left[\begin{array}{c}{t}\\[4pt]{0}\end{array}\right]\)

7. \(-\frac{1}{6}\left[\begin{array}{c}{2-6t}\\[4pt]{7+6t}\\[4pt]{1-12t}\end{array}\right]\)

8. \(-\frac{1}{6}\left[\begin{array}{c}{3e^{t}+4}\\[4pt]{6e^{t}-4}\\[4pt]{10}\end{array}\right]\)

9. \(\frac{1}{18}\left[\begin{array}{c}{e^{t}(1+12t)-e^{-5t}(1+6t)}\\[4pt]{-2e^{t}(1-6t)-e^{-5t}(1-12t)}\\[4pt]{e^{t}(1+12t)-e^{-5t}(1+6t)}\end{array}\right]\)

10. \(\frac{1}{3}\left[\begin{array}{c}{2e^{t}}\\[4pt]{e^{t}}\\[4pt]{2e^{t}}\end{array}\right]\)

11. \(\left[\begin{array}{c}{t\sin t}\\[4pt]{0}\end{array}\right]\)

12. \(-\left[\begin{array}{c}{t^{2}}\\[4pt]{2t}\end{array}\right]\)

13. \((t-1)(\ln |t-1|+t)\left[\begin{array}{c}{1}\\[4pt]{-1}\end{array}\right]\)

14. \(\frac{1}{9}\left[\begin{array}{c}{5e^{2t}-e^{-3t}}\\[4pt]{e^{3t}-5e^{-2t}}\end{array}\right]\)

15. \(\frac{1}{4t}\left[\begin{array}{c}{2t^{3}\ln |t|+t^{3}(t+2)}\\[4pt]{2\ln |t|+3t-2}\end{array}\right]\)

16. \(\frac{1}{2}\left[\begin{array}{c}{te^{-t}(t+2)+(t^{3}-2)}\\[4pt]{te^{t}(t-2)+(t^{3}+2)}\end{array}\right]\)

17. \(-\left[\begin{array}{c}{t}\\[4pt]{t}\\[4pt]{t}\end{array}\right]\)

18. \(\frac{1}{4}\left[\begin{array}{c}{-3e^{t}}\\[4pt]{1}\\[4pt]{e^{-t}}\end{array}\right]\)

19. \(\left[\begin{array}{c}{2t^{2}+t}\\[4pt]{t}\\[4pt]{-t}\end{array}\right]\)

20. \(\frac{e^{t}}{4t}\left[\begin{array}{c}{2t+1}\\[4pt]{2t-1}\\[4pt]{2t+1}\end{array}\right]\)

22.

\({\bf y'}=\left[\begin{array}{cccc}{0}&{1}&{\ldots }&{0}\\[4pt]{0}&{0}&{\ldots }&{0}\\[4pt]{\vdots }&{\vdots }&{\ddots }&{\vdots }\\[4pt]{0}&{0}&{\ldots }&{1}\\[4pt]{-P_{n}(t)/P_{0}(t)}&{-P_{n-1}/P_{0}(t)}&{\ldots }&{-P_{1}(t)/P_{0}(t)}\end{array}\right]{\bf y}+\left[\begin{array}{c}{0}\\[4pt]{0}\\[4pt]{\vdots }\\[4pt]{F(t)/P_{0}(t)}\end{array}\right]\)
\(\left[\begin{array}{cccc}{y_{1}}&{y_{2}}&{\ldots }&{y_{n}}\\[4pt]{y'_{1}}&{y'_{2}}&{\ldots }&{y'_{n}}\\[4pt]{\vdots }&{\vdots }&{\ddots }&{\vdots }\\[4pt]{y_{1}^{(n-1)}}&{y_{2}^{(n-1)}}&{\ldots }&{y_{n}^{(n-1)}}\end{array}\right]\)

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