Section 8.2

1. $x (t) = \frac{7}{3}, y (t) = - \frac{8}{3}$ $x (t) = \frac{7}{3}, y (t) = - \frac{8}{3}$
2. $x (t) = \frac{1}{8} (1 + 12 t), y (t) = - \frac{1}{4} (5 + 4 t)$ $x (t) = \frac{1}{8} (1 + 12 t), y (t) = - \frac{1}{4} (5 + 4 t)$
3. $x (t) = \frac{1}{756} (864 e^{- t} + 4 e^{6 t} - 868 + 840 t - 504 t^{2}), y (t) = \frac{1}{756} (- 864 e^{- t} + 3 e^{6 t} + 861 - 882 t + 378 t^{2})$ $x (t) = \frac{1}{756} (864 e^{- t} + 4 e^{6 t} - 868 + 840 t - 504 t^{2}), y (t) = \frac{1}{756} (- 864 e^{- t} + 3 e^{6 t} + 861 - 882 t + 378 t^{2})$
4. $x (t) = \frac{1}{84} (99 e^{5 t} - 8 e^{- 2 t} - 7 e^{t}), y (t) = \frac{1}{84} (99 e^{5 t} + 48 e^{- 2 t} - 63 e^{t})$ $x (t) = \frac{1}{84} (99 e^{5 t} - 8 e^{- 2 t} - 7 e^{t}), y (t) = \frac{1}{84} (99 e^{5 t} + 48 e^{- 2 t} - 63 e^{t})$
5. $x (t) = \frac{1}{3} (- 12 - e^{- t} - 7 t e^{- t}), y (t) = \frac{1}{3} (- 6 - 7 t e^{- t})$ $x (t) = \frac{1}{3} (- 12 - e^{- t} - 7 t e^{- t}), y (t) = \frac{1}{3} (- 6 - 7 t e^{- t})$
6. $x (t) = - \frac{1}{256} (91 + 16 t) e^{t}, y (t) = \frac{1}{32} (25 + 16 t) e^{t}$ $x (t) = - \frac{1}{256} (91 + 16 t) e^{t}, y (t) = \frac{1}{32} (25 + 16 t) e^{t}$
7. $x (t) = \frac{1}{410} (369 e^{t} + 166 e^{- 9 t} - 125 \cos t - 105 \sin t), y (t) = \frac{1}{410} (369 e^{t} - 249 e^{- 9 t} - 120 \cos t - 150 \sin t)$ $x (t) = \frac{1}{410} (369 e^{t} + 166 e^{- 9 t} - 125 \cos t - 105 \sin t), y (t) = \frac{1}{410} (369 e^{t} - 249 e^{- 9 t} - 120 \cos t - 150 \sin t)$
8. $x (t) = \frac{1}{3} (17 \cos t + 2 \sin t), y (t) = \frac{1}{3} (3 \cos t + 5 \sin t)$ $x (t) = \frac{1}{3} (17 \cos t + 2 \sin t), y (t) = \frac{1}{3} (3 \cos t + 5 \sin t)$
9. $x (t) = \frac{1}{4} (\sin 2 t + 2 t \cos 2 t + t \sin 2 t), y (t) = \frac{1}{4} t \sin 2 t$ $x (t) = \frac{1}{4} (\sin 2 t + 2 t \cos 2 t + t \sin 2 t), y (t) = \frac{1}{4} t \sin 2 t$
10. $x (t) = \frac{1}{13} e^{t} (4 \cos t - 6 \sin t), y (t) = \frac{1}{13} e^{t} (3 \cos t + 2 \sin t)$ $x (t) = \frac{1}{13} e^{t} (4 \cos t - 6 \sin t), y (t) = \frac{1}{13} e^{t} (3 \cos t + 2 \sin t)$
11. $x (t) = \frac{1}{2} (1 - 4 t + e^{4 t}), y (t) = \frac{1}{4} (- 5 + 4 t + e^{4 t})$ $x (t) = \frac{1}{2} (1 - 4 t + e^{4 t}), y (t) = \frac{1}{4} (- 5 + 4 t + e^{4 t})$
12. $x (t) = t^{2}, y (t) = - t^{2}$ $x (t) = t^{2}, y (t) = - t^{2}$
13. $x (t) = \frac{1}{2} (1 + 5 t) e^{t}, y (t) = - \frac{5}{2} t e^{t}$ $x (t) = \frac{1}{2} (1 + 5 t) e^{t}, y (t) = - \frac{5}{2} t e^{t}$
14. $x (t) = \frac{1}{8} (- 2 + 4 t - e^{4 t} + 2 {t e}^{4 t}), y (t) = \frac{1}{2} t (- 2 + e^{4 t})$ $x (t) = \frac{1}{8} (- 2 + 4 t - e^{4 t} + 2 {t e}^{4 t}), y (t) = \frac{1}{2} t (- 2 + e^{4 t})$
15. (a) $x_{1} (t) = 200 (1 - e^{- t / 10}), x_{2} (t) = 400 (1 + e^{- t / 10} - 2 e^{- t / 20}) (b) x_{1} (t) \to 200$ $x_{1} (t) = 200 (1 - e^{- t / 10}), x_{2} (t) = 400 (1 + e^{- t / 10} - 2 e^{- t / 20}) (b) x_{1} (t) \to 200$ and $x_{2} (t) \to 400$ $x_{2} (t) \to 400$ as $t \to + \infty (c)$ $t \to + \infty (c)$ Tank 1: about 6 min 56 s; tank 2: about 24 min 34 s
16. (a) $x_{1} (t) = 600 (1 - e^{- t / 20}), x_{2} (t) = 300 (1 + e^{- t / 10} - 2 e^{- t / 20})$ $x_{1} (t) = 600 (1 - e^{- t / 20}), x_{2} (t) = 300 (1 + e^{- t / 10} - 2 e^{- t / 20})$ (b) $x_{1} (t) \to 600$ $x_{1} (t) \to 600$ and $x_{2} (t) \to 300$ $x_{2} (t) \to 300$ as $t \to \infty$ $t \to \infty$ (c) Tank 1: about 8 min 7 sec; tank 2: about 17 min 13 sec
17. $x_{1} (t) = 102 - 95 e^{- t} - 7 e^{5 t}, x_{2} (t) = 96 - 95 e^{- t} - e^{5 t}$ $x_{1} (t) = 102 - 95 e^{- t} - 7 e^{5 t}, x_{2} (t) = 96 - 95 e^{- t} - e^{5 t}$
18. $x_{1} (t) = 68 - 110 t - 75 e^{- t} + 7 e^{5 t}, x_{2} (t) = 74 - 80 t - 75 e^{- t} + e^{5 t}$ $x_{1} (t) = 68 - 110 t - 75 e^{- t} + 7 e^{5 t}, x_{2} (t) = 74 - 80 t - 75 e^{- t} + e^{5 t}$
19. $x_{1} (t) = - 70 - 60 t + 16 e^{- 3 t} + 54 e^{2 t}, x_{2} (t) = 5 - 60 t - 32 e^{- 3 t} + 27 e^{2 t}$ $x_{1} (t) = - 70 - 60 t + 16 e^{- 3 t} + 54 e^{2 t}, x_{2} (t) = 5 - 60 t - 32 e^{- 3 t} + 27 e^{2 t}$
20. $x_{1} (t) = 3 e^{2 t} + 60 {t e}^{2 t} - 3 e^{- 3 t}, x_{2} (t) = - 6 e^{2 t} + 30 {t e}^{2 t} + 6 e^{- 3 t}$ $x_{1} (t) = 3 e^{2 t} + 60 {t e}^{2 t} - 3 e^{- 3 t}, x_{2} (t) = - 6 e^{2 t} + 30 {t e}^{2 t} + 6 e^{- 3 t}$
21. $x_{1} (t) = - e^{- t} - 14 e^{2 t} + 15 e^{3 t}, x_{2} (t) = - 5 e^{- t} - 10 e^{2 t} + 15 e^{3 t}$ $x_{1} (t) = - e^{- t} - 14 e^{2 t} + 15 e^{3 t}, x_{2} (t) = - 5 e^{- t} - 10 e^{2 t} + 15 e^{3 t}$
22. $x_{1} (t) = - 10 e^{- t} - 7 {t e}^{- t} + 10 e^{3 t} - 5 {t e}^{3 t},$ $x_{1} (t) = - 10 e^{- t} - 7 {t e}^{- t} + 10 e^{3 t} - 5 {t e}^{3 t},$ $x_{2} (t) = - 15 e^{- t} - 35 {t e}^{- t} + 15 e^{3 t} - 5 {t e}^{3 t}$ $x_{2} (t) = - 15 e^{- t} - 35 {t e}^{- t} + 15 e^{3 t} - 5 {t e}^{3 t}$
23. $x_{1} (t) = 3 + 11 t + 8 t^{2}, x_{2} (t) = 5 + 17 t + 24 t^{2}$ $x_{1} (t) = 3 + 11 t + 8 t^{2}, x_{2} (t) = 5 + 17 t + 24 t^{2}$
24. $x_{1} (t) = 2 + t + \ln t, x_{2} (t) = 5 + 3 t - \frac{1}{t} + 3 \ln t$ $x_{1} (t) = 2 + t + \ln t, x_{2} (t) = 5 + 3 t - \frac{1}{t} + 3 \ln t$
25. $x_{1} (t) = - 1 + 8 t + \cos t - 8 \sin t, x_{2} (t) = - 2 + 4 t + 2 \cos t - 3 \sin t$ $x_{1} (t) = - 1 + 8 t + \cos t - 8 \sin t, x_{2} (t) = - 2 + 4 t + 2 \cos t - 3 \sin t$
26. $x_{1} (t) = 3 \cos t - 32 \sin t + 17 t \cos t + 4 t \sin t,$ $x_{1} (t) = 3 \cos t - 32 \sin t + 17 t \cos t + 4 t \sin t,$ $x_{2} (t) = 5 \cos t - 13 \sin t + 6 t \cos t + 5 t \sin t$ $x_{2} (t) = 5 \cos t - 13 \sin t + 6 t \cos t + 5 t \sin t$
27. $x_{1} (t) = 8 t^{3} + 6 t^{4}, x_{2} (t) = 3 t^{2} - 2 t^{3} + 3 t^{4}$ $x_{1} (t) = 8 t^{3} + 6 t^{4}, x_{2} (t) = 3 t^{2} - 2 t^{3} + 3 t^{4}$
28. $x_{1} (t) = - 7 + 14 t - 6 t^{2} + 4 t^{2} \ln t,$ $x_{1} (t) = - 7 + 14 t - 6 t^{2} + 4 t^{2} \ln t,$ $x_{2} (t) = - 7 + 9 t - 3 t^{2} + \ln t - 2 t \ln t + 2 t^{2} \ln t$ $x_{2} (t) = - 7 + 9 t - 3 t^{2} + \ln t - 2 t \ln t + 2 t^{2} \ln t$
29. $x_{1} (t) = t \cos t - (\ln \cos t) (\sin t), x_{2} (t) = t \sin t + (\ln \cos t) (\cos t)$ $x_{1} (t) = t \cos t - (\ln \cos t) (\sin t), x_{2} (t) = t \sin t + (\ln \cos t) (\cos t)$
30. $x_{1} (t) = \frac{1}{2} t^{2} \cos 2 t, x_{2} (t) = \frac{1}{2} t^{2} \sin 2 t$ $x_{1} (t) = \frac{1}{2} t^{2} \cos 2 t, x_{2} (t) = \frac{1}{2} t^{2} \sin 2 t$
31. $x_{1} (t) = (9 t^{2} + 4 t^{3}) e^{t}, x_{2} (t) = 6 t^{2} e^{t}, x_{3} (t) = 6 t e^{t}$ $x_{1} (t) = (9 t^{2} + 4 t^{3}) e^{t}, x_{2} (t) = 6 t^{2} e^{t}, x_{3} (t) = 6 t e^{t}$
32. $x_{1} (t) = (44 + 18 t) e^{t} + (- 44 + 26 t) e^{2 t},$ $x_{1} (t) = (44 + 18 t) e^{t} + (- 44 + 26 t) e^{2 t},$ $x_{2} (t) = 6 e^{t} + (- 6 + 6 t) e^{2 t}, x_{3} (t) = 2 {t e}^{2 t}$ $x_{2} (t) = 6 e^{t} + (- 6 + 6 t) e^{2 t}, x_{3} (t) = 2 {t e}^{2 t}$
33. $x_{1} (t) = 15 t^{2} + 60 t^{3} + 95 t^{4} + 12 t^{5}, x_{2} (t) = 15 t^{2} + 55 t^{3} + 15 t^{4}, x_{3} (t) = 15 t^{2} + 20 t^{3}, x_{4} (t) = 15 t^{2}$ $x_{1} (t) = 15 t^{2} + 60 t^{3} + 95 t^{4} + 12 t^{5}, x_{2} (t) = 15 t^{2} + 55 t^{3} + 15 t^{4}, x_{3} (t) = 15 t^{2} + 20 t^{3}, x_{4} (t) = 15 t^{2}$
34. $x_{1} (t) = 4 t^{3} + (4 + 16 t + 8 t^{2}) e^{2 t}, x_{2} (t) = 3 t^{2} + (2 + 4 t) e^{2 t}, x_{3} (t) = (2 + 4 t + 2 t^{2}) e^{2 t}, x_{4} (t) = (1 + t) e^{2 t}$ $x_{1} (t) = 4 t^{3} + (4 + 16 t + 8 t^{2}) e^{2 t}, x_{2} (t) = 3 t^{2} + (2 + 4 t) e^{2 t}, x_{3} (t) = (2 + 4 t + 2 t^{2}) e^{2 t}, x_{4} (t) = (1 + t) e^{2 t}$

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Table of Contents for Section 8.2

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Table of Contents for
Section 8.2