Let c be a positive number. A differential equation of the form dy/dt=ky^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. An especially prolific breed of rabbits has the growth term My^1.01. If 2 such rabbits breed initially and the warren has 16 rabbits after three months, then when is doomsday?

The doomsday is 146 days

Step-by-step explanation:

Given

First, we calculate the solution that satisfies the initial solution

Multiply both sides by

Take integral of both sides

Integrate

To find c; let t= 0

Substitute in

Multiply through by -c

Take exponents of

i.e.

Next:

i.e. 3 months

--- initial number of breeds

So, we have:

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We have the growth term to be:

This implies that:

By comparison:

--- 16 rabbits after 3 months:

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Take -1/100th root of both sides

Recall that:

This implies that:

Make T the subject

Substitute: , and

The doomsday is 146 days