Chapter 9.4, Problem 18E

Let c be a positive number. A differential equation of the form

$\frac{dy}{dt}=k{y}^{1+c}$

where k is a positive constant, is called a doomsday equation because the exponent in the expression ky^1+c is larger than the exponent 1 for natural growth.

1. (a) Determine the solution that satisfies the initial condition y(0) = y₀.
2. (b) Show that there is a finite time t = T (doomsday) such that lim_t→T − y(t) = ∞.
3. (c) An especially prolific breed of rabbits has the growth term ky^1.01. If 2 such rabbits breed initially and the warren has 16 rabbits after three months, then when is doomsday?