We have a population model given by the differential equation:
dP/dt = (k*cos(t))*P, where k is a pos. constant, and P(T) undergoes yearly seasonal changes.
Assume that P(0) = P_(0).
1.) Solve the Dif. Equation and graph the solution.
Did you notice that the differential equation is separable?
$\displaystyle \frac{dP}{dt} = (k~cos(t))P$
$\displaystyle \frac{dP}{P} = k~cos(t)~dt$
$\displaystyle \int \frac{dP}{P} = \int k~cos(t)~dt$
So integrate both sides and use your initial condition to evaluate the arbitrary constant in terms of $\displaystyle P_0$.
-Dan