Solving the Doomsday Equation: Initial Conditions, Finite Time & Rabbit Warrens

[noboost4you]

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 equation in the expression ky^(1+c) is larger than that for natural growth (that is, ky).

(a) Determine the solution that satisfies the initial condition y(0)=y(subzero)
(b) Show that there is a finite time t = T (doomsday) such that lim(t->T-) y(t) = infinity
(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?

We just got finished learning Radioactive Decay and Newton's Law of Cooling sections which this question has come from and I have no idea even how to approach this such question.

Any help would be appreciated. Don't flat out give me the answer, but offer any positive assistance. Thanks!

 

[cookiemonster]

Can you rearrange the equation to get only t's (and dt's) on one side and only y's (and dy's) on the other? If there are only t's on one side and y's on the other, can you think of a way to get rid of the differentials?

cookiemonster

 

[HallsofIvy]

How much of hint do you need? That's a separable equation (that's what cookiemonster was telling you) and can be written as
$$\frac{dy}{y^{1+c}}= kdt$$
which can be integrated.


By the way it's called the "doomsday" equation not just "because the equation in the expression ky^(1+c) is larger than that for natural growth (that is, ky)" but because there is a singularity: at some finite time (t= y₀/(ck)) the population goes to infinity: "doomsday".