Homework Two

10

$$y(x) = \pm \sqrt{\frac{2}{3}x^3 + 2 C}.$$

11

$$y(x) = - \frac{1}{\cos x +C},$$

also by inspection

$$y(x)=0.$$

12

$$y(x) = \sin\left[ \log\vert x\vert + C \right],$$

also by inspection,

$$y(x)=\pm 1.$$

13

$$y(x) = - \sqrt{\frac{x^2 + 1}{2}},$$

defined for

$$-\infty<x<\infty.$$

14

$$y(x) = - \sqrt{\frac{2}{3}\log\vert 1+x^3\vert+1}.$$

15

$$\tau = \frac{1}{r}\log{2} = 20.0 {\rm days.}$$

16

(a)

$$Q(t) = 35.36 + 64.64 e^{ - 0.02828 t}.$$

(b)

$$\lim\limits_{t\to\infty} = 35.36 {\rm mg}.$$

(c)

$$T = 171.9 {\rm days}$$

(d)

$$k = r \cdot 100{\rm mg} = 2.828 {\rm mg}.$$

17

$$Q(t) = 120 \gamma (1 - e^{-\frac{t}{60}}),$$

limiting amount is

$$120 \gamma.$$

18

(a)

$$x(t) = c ( 1 - e^{-\frac{\phi}{V} t}) = 0.04 (1-e^{-\frac{t}{13000}}).$$

(b)

$$\tau = \frac{V}{\Phi}\log\frac{1}{1-\frac{x(\tau)}{c}} = 36 {\rm min.}$$