How do I solve this problem?

Question

Let c be a positive number. A differential equation of the form below 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. An especially prolific breed of rabbits has the growth term ky^1.01. If 2 such rabbits breed initially and the warren has 29 rabbits after three months, then when is doomsday? (Doomsday is the finite time t=T such that . Round the answer to two decimal places.)

Answer 1

STARTING FROM YOUR EQUATION: dy/dt = ky^1.01

1. separate the variables
2. integrate both sides; generating a constant c on the right
3. get rid of the 1.01 by changing c and k to C and K , respectively
4. you have a fresh equation in terms of y, t, C and K
5. get C by plugging in (t= 0,y= 2)the first condition
6. get K by plugging in (t = 3, y = 29 and C from (5))
7. now you have an equation that you can solve for y(t); t is the only variable
8. examine the limit of right side that equation as t->inf
9. Doomsday, T, is the floor of the t that makes the right side of that equation increase without bound.