Differential Equations and Linear Algebra, 4th Edition

Section 5.4

1. 1. Frequency: 2 rad/s ($1/\pi \text{ HZ});$ period: $\pi \text{ s}$

2. 2. Frequency: 8 rad/sec ($4/\pi \text{ HZ});$ period: $\pi /4\text{ sec}$

3. 3. Amplitude: 2m; frequency: 5 rad/s; period: $2\pi /5\text{ s}$

4. 4. (a) $x(t)=\frac{13}{12}\cos (12t-\alpha )$ with $\alpha =2\pi -{\tan }^{-1}(5/12)\approx 5.8884$; (b) Amplitude: $\frac{13}{12}$ m; period: $\pi /6\text{ sec}$

5. 6. About 7.33 mi

6. 7. About 10450 ft

7. 8. 29.59 in.

8. 10. Amplitude: 100 cm; period: about 2.01 sec

9. 11. About 3.8 in.

10. 13. (a) $x(t)=50(e^{-2t/5}-e^{-t/2})$; (b) 4.096 exactly

11. 14. (a) $x(t)=25e^{-t/5}\cos (3t-\alpha )$ with $\alpha ={\tan }^{-1}(3/4)\approx 0.6435$; (b) envelope curves $x=\pm 25e^{-t/5}$; pseudoperiod $2\pi /3$

12. 15. $x(t)=4e^{-2t}-2e^{-4t}$, overdamped; $u(t)=2\cos (2\sqrt{2}\hspace{1em}t)$

    

13. 16. $x(t)=4e^{-3t}-2e^{-7t}$, overdamped; $u(t)\approx 2\sqrt{\frac{22}{21}}\cos (\sqrt{21}\hspace{1em}t-0.2149)$

14. 17. $x($