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

[tex]\frac{dy}{dt} = ky^{1+c}[/tex]

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

(a) Determine the solution that satisfies the condition [tex]y(0) = y_{0}[/tex]
(b) Show that there is a finite time [tex]t = T[/tex] (doomsday) such that [tex]\lim_{t \to T^{-}} y(t) = \infty[/tex]
(c) An especially prolific breed of rabbits has the growth term [tex]ky^{1.01}[/tex]. If 2 such rabbits breed initially and the warren has 16 rabbits after three months, when is the doomsday?