# Differential Equations

• Jun 18th 2007, 07:43 AM
classicstrings
Differential Equations
A model for the number W of wombats on a island, t years after a initial 200 are settled there, takes into account the availability of wombat food. Under this model, dW/dt = (m - n - kW)W where m and n are related to the birth rate and death rate respectively, and k is a positive constant related to the amount of food available on the island.

Suppose m = 0.10, n = 0.06 and k = 0.00005 and W < 800. Find an expression for the number of wombats on the island after t years. Find a general solution for the differential equation that represents the model for W > ((m-n)/k).

What happens as t approaches infinity? Use calculus to find at what time, t, is the wombat population increasing most rapidly.

...

I've tried to antidiff, by flipping to get dt/dw and then t = (1/(m-n-kw))logeW + c. I have to find a C value to find a general solution, and I get (t=0, w=200), c = (loge200 - 0.01)/(-0.04). Then I get stuck on simplifying to get a general solution.
• Jun 18th 2007, 10:56 AM
topsquark
Quote:

Originally Posted by classicstrings
A model for the number W of wombats on a island, t years after a initial 200 are settled there, takes into account the availability of wombat food. Under this model, dW/dt = (m - n - kW)W where m and n are related to the birth rate and death rate respectively, and k is a positive constant related to the amount of food available on the island.

Suppose m = 0.10, n = 0.06 and k = 0.00005 and W < 800. Find an expression for the number of wombats on the island after t years. Find a general solution for the differential equation that represents the model for W > ((m-n)/k).

What happens as t approaches infinity? Use calculus to find at what time, t, is the wombat population increasing most rapidly.

...

I've tried to antidiff, by flipping to get dt/dw and then t = (1/(m-n-kw))logeW + c. I have to find a C value to find a general solution, and I get (t=0, w=200), c = (loge200 - 0.01)/(-0.04). Then I get stuck on simplifying to get a general solution.

$\frac{dW}{dt} + (n - m)W = -kW^2$

is a Bernoulli differential equation. (They are of the form: $y ^{\prime} + P(x)y = Q(x)y^n$ where n is an integer.)

Use the substitution:
$W(t) = \frac{1}{y(t)}$

Then
$\frac{dW}{dt} = -\frac{1}{y^2} \frac{dy}{dt}$

and
$-\frac{1}{y^2} \frac{dy}{dt} + \frac{(n - m)}{y} = \frac{-k^2}{y^2}$

or
$\frac{dy}{dt} - (n - m)y = k^2$

which I'm sure you can solve from here.

-Dan