22: Section 6.2 Answers

1. \(y=\frac{e^{-2t}}{2}(3\cos 2t-\sin 2t)\text{ ft};\:\sqrt{\frac{5}{2}}e^{-2t}\text{ ft}\)

2. \(y=-e^{-t}\left(3\cos 3t+\frac{1}{3}\sin 3t \right)\text{ ft}\frac{\sqrt{82}}{3}e^{-t}\text{ ft}\)

3. \(y=e^{-16t}\left(\frac{1}{4}+10t \right)\text{ ft}\)

4. \(y=-\frac{e^{-3t}}{4}(5\cos t+63\sin t)\text{ ft}\)

5. \(0\leq c<8\text{ lb-sec/ft}\)

6. \(y=\frac{1}{2}e^{-3t}\left(\cos\sqrt{91}t+\frac{11}{\sqrt{91}}\sin\sqrt{91}t \right)\text{ ft}\)

7. \(y=-\frac{e^{-4t}}{3}(2+8t)\text{ ft}\)

8. \(y=e^{-10t}\left(9\cos 4\sqrt{6}t+\frac{45}{2\sqrt{6}}\sin 4\sqrt{6}t \right)\text{ cm}\)

9. \(y=e^{-3t/2}\left(\frac{3}{2}\cos\frac{\sqrt{41}}{2}t+\frac{9}{2\sqrt{41}}\sin\frac{\sqrt{41}}{2}t \right)\text{ ft}\)

10. \(y=e^{-\frac{3}{2}t}\left(\frac{1}{2}\cos\frac{\sqrt{119}}{2}t-\frac{9}{2\sqrt{119}}\sin\frac{\sqrt{119}}{2}t \right)\text{ ft}\)

11. \(y=e^{-8t}\left(\frac{1}{4}\cos 8\sqrt{2}t-\frac{1}{4\sqrt{2}}\sin 8\sqrt{2}t \right)\text{ ft}\)

12. \(y=e^{-t}\left(-\frac{1}{3}\cos 3\sqrt{11}t+\frac{14}{9\sqrt{11}}\sin 3\sqrt{11}t \right)\text{ ft}\)

13. \(y_{p}=\frac{22}{61}\cos 2t+\frac{2}{61}\sin 2t\text{ ft}\)

14. \(y=-\frac{2}{3}(e^{-8t}-2e^{-4t})\)

15. \(y=e^{-2t}\left(\frac{1}{10}\cos 4t-\frac{1}{5}\sin 4t \right)\text{ m}\)

16. \(y=e^{-3t}(10\cos t-70\sin t)\text{ cm}\)

17. \(y_{p}=-\frac{2}{15}\cos 3t+\frac{1}{15}\sin 3t\text{ ft}\)

18. \(y_{p}=\frac{11}{100}\cos 4t+\frac{27}{100}\sin 4t\text{ cm}\)

19. \(y_{p}=\frac{42}{73}\cos t+\frac{39}{73}\sin t\text{ ft}\)

20. \(y=-\frac{1}{2}\cos 2t+\frac{1}{4}\sin 2t\text{ m}\)

21. \(y_{p}=\frac{1}{c\omega _{0}}(-\beta\cos\omega _{0}t+\alpha\sin\omega _{0}t)\)

24. a. \(t=1/4\) s
     b. \(t=1/2\) s and \(y=-e^{-2}\) ft

25. \(y=-{4\over 3}e^{-2t}+{1\over 3}e^{-8t}\)

26. \(y={2\over 3}e^{-2t}-{5\over 3}e^{-8t}\)

27. a. \(y=e^{-2t}\cos t+2e^{-2t}\sin t\)
     b. \(y=\sqrt{5}e^{-2t}\sin (t+\tan^{-1} {1\over 2})\)
     c. \(t=n\pi-\tan^{-1}{1\over 2}\)
     d. \(t=2\pi-\tan^{-1} {1\over 2}\) s

28. \(y=-{16\over 3}e^{-t/2}\cos {\sqrt{47}\over 2}t-{76\over 3\sqrt{47}}e^{-t/2}\sin {\sqrt{47}\over 2}t+{10\over 3}\cos 3t+{10\over 3}\sin 3t\)

29. \(y={1\over 4}e^{-4t}+te^{-4t}-{1\over 4}\cos 4t\)

30. \(y={1\over 2}\cos 4t-{9\over 4}\sin 4t-{1\over 2}e^{-2t}\cos 4t+2e^{-2t}\sin 4t\)

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