Consider the differential equation

dP/dt = kP^1+c,

where k > 0 and c ≥ 0. In Section 3.1 we saw that in thecase c = 0 the linear differential equation dP/dt = kP is a mathematical model of a population P(t) that exhibits unbounded growth over the in­finite time interval [0, ∞), that is, P(t) → ∞ as t → ∞. See Example 1 on page 85.

(a) Suppose for c = 0.01 that the nonlinear differential equation

dP/dt = kP^1.01, k > 0,

is a mathematical model for a population of small animals, where time t is measured in months. Solve the differential equation subject to the initial condition P(0) = 10 and the fact that the animal population has doubled in 5months.

(b) The differential equation in part (a) is called a doomsday equation because the population P(t) exhibits unbounded growth over a fi­nite time interval (0, T), that is, there is some time T such that P(t) → as t → T^-. Find T.

(c) From part (a), what is P(50)? P(100)?