Homework Three

19 $N=0$ (stable), $N=-a/b$ (unstable)

20 $N=0$ (unstable), $N=1$ (stable), $N=2$ (unstable).

21

(a) The only equilibrium is $N=0$,

(c)

$$N=\frac{N_0 + (1-N_0) k t}{1+(1-N_0)k t}.$$

22 $N=-1$ (stable), $N=0$ (unstable), $N=1$ (stable).

23 $N=-2$ (unstable), $N=0$ (semistable), $N=2$ (stable).

24

(a)

$$N_{12} = \frac{k}{2}\pm\sqrt{\frac{k^2}{4} - \frac{k h}{r}},$$

where $4 h < k R$ implies real positive discriminant.

(b) $N^2$ has negative value, parabola has a maximum, left root is unstable, right is stable.

25

$$y(x) = C_1 e^{-3x} + C_2 e^x.$$

26

$$y(x) = C_1 + C_2 e^{-5x}.$$

27

$$y(x) = C_1 e^{ (1+\sqrt{3})x} + C_2 e^{(1-\sqrt{3})x}.$$

28

$$y(x) = C_1 e^{-9x} + C_2e^x.$$

$$y(x) = \frac{1}{10} e^{-9x} + \frac{9}{10}e^x.$$

29

$$y = C_1 e^x \cos \sqrt{5} x + C_2 e^x \sin \sqrt{5}{x}.$$

30

$$y(x) = C_1 e^{-x} \cos{x} + C_2 s^{-x} \sin {x}.$$

31

$$y(x) = C_1 e^{-3x}\cos 2 x + C_2 s^{-3x} \sin 2 x.$$

32

$$y(x) = C_1 e^{-2 x} \cos x + C_2 e^{-2 x} \sin x,$$

$$C_2 = 2 C_1 = 2.$$

33

$$y(x) = C_1 e^x + C_2 x e^x.$$

34

$$y(x) = C_1 e^{2x} + C_2 x e^{3x}.$$