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.