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

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

where k is a positive constant, is called a doomsday equation because the exponent in the expression ky1+c is larger than the exponent 1 for natural growth.

(a) Determine the solution that satisfies the initial condition y(0) = y0

(b) Show that there is a finite time t = T (doomsday) such that limt→T–y(t) = ∞.

(c) An especially prolific breed of rabbits has the growth term ky1.01. If 2 such rabbits breed initially and the warren has 16 rabbits after three months, then when is doomsday?

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