In the preceding problem, a qualitative analysis of the differential equation in (1.5.7)...

In the preceding problem, a qualitative analysis of the differential equation in (1.5.7) was carried out. In this problem, we determine the exact solution to the differential equation and verify the predictions from the qualitative analysis.

(a) Solve the initial-value problem (1.5.7).

(b) Using your solution from (a), verify that if P0 < T , then . What does this mean for the population?

(c) Using your solution from (a), verify that if P0 > T , then each solution curve has a vertical asymptote at t = t_e, where

How do you interpret this result in terms of population growth? Note that this was not obvious from the qualitative analysis performed in the previous problem.