91Ó°ÊÓ

Find and interpret all equilibrium points for the predator-prey model. $$\left\\{\begin{array}{l}x^{\prime}=0.2 x-0.1 x^{2}-0.4 x y \\\ y^{\prime}=-0.3 y+0.1 x y\end{array}\right.$$

Involve exponential growth. Suppose a bacterial culture initially has 400 cells. After 1 hour, the population has increased to \(800 .\) Find an equation for the population at any time. What will the population be after 10 hours?

Involve exponential growth. Suppose a bacterial culture initially has 100 cells. After 2 hours, the population has increased to \(400 .\) Find an equation for the population at any time. What will the population be after 8 hours?

Involve exponential decay. The half-life of morphine in the human bloodstream is 3 hours. If initially there is 0.4 mg of morphine in the bloodstream, find an equation for the amount in the bloodstream at any time. When does the amount drop below \(0.01 \mathrm{mg} ?\)

The differential equation is separable. Find the general solution, in an explicit form if possible. Sketch several members of the family of solutions. $$y^{\prime}=\frac{\left(y^{2}+1\right) \ln x}{4 y}$$

Involve Newton's Law of Cooling. At 10: 07 P.M. you find a secret agent murdered. Next to him is a martini that got shaken before the secret agent could stir it. Room temperature is \(70^{\circ} \mathrm{F}\). The martini warms from \(60^{\circ} \mathrm{F}\) to \(61^{\circ} \mathrm{F}\) in the 2 minutes from 10: 07 P.M. to 10: 09 P.M. If the secret agent's martinis are always served at \(40^{\circ} \mathrm{F}\), what was the time of death?