Numerical Solutions of Differential Equations

Abstract

The reader is probably familiar with the differential equation y′ = ky from calculus. Often it is introduced as a simple model for population growth or accumulating interest. For example, if p(x) represents the population of bacteria in a laboratory sample at time x and the population increases by fixed percentage during each time interval, then it is reasonable to assume that there is k > 0 so that p′ = kp: the rate of change of the size of the population, as reflected by the derivative, is proportional to the population. The larger the population, the more individuals that are added to it during each time interval. If we wish a more sophisticated model that takes into account the fact that the crowding of a large population may effect growth negatively, then we might reflect this by adding a quadratic term to get w′ = kw − aw² where both k and a are positive. When the population is small, kw is greater than aw², the derivative is positive, and the population is growing. As w increases the difference between kw and aw² decreases until we reach a point where aw² is larger than kw, the derivative is negative and the population is shrinking.