Homework Six

58

$$R=5, \tan \delta = \frac{4}{3}.$$

59

$$R=2, \delta = -\frac{\Pi}{3}.$$

60

$$U = -\frac{1}{8\sqrt{3}}e^{-2 t} \sin (2\sqrt{3})$$

$$T = \frac{\pi}{2\sqrt{3}}{\rm sec}.$$

$$\tau = \frac{1}{2}\ln\frac{150}{\sqrt{31}}= 1.59 { sec}.$$

61

$$u(t) = \sqrt{2}\sin\sqrt{2}t.$$

$u$, $\dot u$ is an ellipse with clockwise rotations

62

$$u(t) = \frac{16}{\sqrt{127}}e^{-\frac{t}{2}}\sin\left(\frac{\sqrt{127}}{8}t\right).$$

$u$, $\dot u$ is an inward going spiral with clock wise rotations, use computer to plot this curve.

63

(a)

$$\ddot u + 10 \dot u + 100 u = 2 \sin \frac{t}{2}.$$

$$u = \frac{1}{250\sqrt{3}}e^{-2 t} \sin 5\sqrt{3}t + \frac{1}{... ... + \frac{1}{50}\sin\frac{t}{2} - \frac{1}{1000}\cos\frac{t}{2}.$$

(b)

Terms with $e^{-2 t}$ are transient, others are steady.

(d)

$$\omega = 5\sqrt{2}.$$

64

$$u = A \cos \omega t + B \sin \omega t,$$

$$A = \frac{2(2-\omega^2)}{(2-\omega^2)^2 + (\frac{\omega}{4})^2},$$

$$B = \frac{\omega}{2}\frac{1}{(2-\omega^2)^2 + (\frac{\omega}{4})^2}.$$

$$R = \frac{2}{\sqrt{(2-\omega^2)^2 + (\frac{\omega}{4})^2}}/$$

$$\omega_{\rm max} = \frac{3\sqrt{14}}{8},$$

$$R_{\rm max} = \frac{64}{\sqrt{127}}.$$