Consider the growth of a population $p(t)$. It starts out with $p(0) = A$. Suppose the growth is unchecked, and hence $$p' = k p$$ for some constant $k$.

Then $p(t) =$ .

Of course populations don't grow forever. Let's say there is a stable population size $Q$ that $p(t)$ approaches as time passes. Thus the speed at which the population is growing will approach zero as the population size approaches $Q$. One way to model this is via the differential equation $$p' = kp(Q-p),\quad p(0) = A.$$ The solution of this initial value problem is $p(t) =$ .
Hint: Proceed as in the last problem.