Introduction

To indicate why the study of differential equations can be useful some simple examples will be considered. The way that the size of a population varies in time is a matter of interest

in several contexts. Let the number of individuals in a given area at time t be p(t). At time t+T the number of individuals is p(t+T ) so that p(t+T )−p(t) must be the number of individuals that have been added to the population during the time interval T . The longer the interval the more individuals can be expected to arrive and the shorter the time the fewer additions can be expected. So write the change in the time interval T as NT and then

p(t+ T )− p(t) = NT

or p(t+ T )− p(t)

T = N.