Chapter 2.3, Problem 17P

Suppose that a certain population has growth rate that varies with time and that this population satisfies the differential equation $\frac{dy}{dt}=\frac{\left(0.5+\sin t\right)y}{5}$.

If $y(0)=1$, find (or estimate) the time $\tau$ at which the population doubled. Choose other initial conditions and determine whether the doubling time $\tau$ depends on the initial population.

Suppose that the growth rate replaced by its average value $\frac{1}{10}$. Determine the doubling time $\tau$ in this case.

Suppose that the term $\sin t$ in the differential equation is replaced by $\sin 2\pi t$; that is, the variation in the growth ratehas a substantially higher frequency. What effect does this have on the doubling time $\tau$?

Plot the solutions obtained in parts (a), (b) and (c) on a single set of axes.