A strain of bacteria in a medium is destroyed by a toxin at a rate jointly proportional to the population of the bacteria and to the amount of the toxin. If there were no toxins, the bacteria would have a growth rate proportional to the amount of bacteria present. Let be the population of bacteria at time , and assume the amount of the toxin is increasing at a constant rate starting with 0 toxins at . Find a differential equation for and solve this equation.
From the problem statement we have that satisfies the differential equation
with . This is a linear first-order differential equation so we may use Theorem 8.3 (page 310 of Apostol) with
The correct curve (see the image on page 322 of Apostol for the curves) is (d) since
Furthermore, is never 0. Finally, we know there is initial growth in at time since is positive.