Problem

The population of fish in a pond is modeled by the differential equation

where time t is measured in years.

Towards what number does the population of fish tend? If there are initially 10 fish in the pond, how long does it take for the number of fish to reach 90% of the eventual population?

Solution

Finding the eventual population is, in math terms, finding

So, we solve the differential equation for the function P, using separation of variables.

and, no matter what the value of C₄,

To solve for C₄, remember that at time 0 there are 10 fish in the pond.

Therefore, the formula for the fish population is

90% of the eventual population of 120 is 108 fish, so to find out at what time the population is reached we solve for t.

So the fish population reaches 108 in a little more than half a year.