A population of bacteria in a Petri dish is modeled by $P^{\prime }\left(t\right)+aP\left(t\right)=S$, where $P\left(t\right)$ is the population at time $t$ (in hours), $a$ is the death rate constant, and $S$ is the supply rate. If $S=40$ cells per hour and $a=0.2$ per hour, what is the equilibrium population?

Consider the differential equation $x^{\prime }\left(t\right)=x(x-2)(5-x)$. In which regions are the solutions decreasing?

Consider the differential equation $y^{\prime }\left(t\right)=\sin y$ for $\mid y\mid \le \frac{\pi }{2}$. Which initial conditions $y\left(0\right)=B$ lead to solutions that are increasing in time?

Sketch a direction field for the following differential equation and the solution curve corresponding to the given initial condition.

$y^{\prime }\left(t\right)=5-y$, $y\left(0\right)=7$

Sketch a direction field for the following differential equation and the solution curve corresponding to the given initial condition. Assume $t\ge 0$ and $P\ge 0$.

$P^{\prime }\left(t\right)=0.02P(1-\frac{P}{1000})$, $P\left(0\right)=200$