TABLE OF CONTENTS Before we start our formal treatment of the subject we will look at a very simple example that nonetheless exhibits the power of differential equations as models of reality. One point to bear in mind in this chapter is the distinction to be made between finding the solution of a differential equation, and interpreting this solution.
Let N( t) denote the number of radioactive atoms in some sample of material at time t. Then with k < 0 the equation
is a very good model for the way that the number of radioactive atoms decays (see Exercise 1.1).
Although we will see later how to solve this equation, for now we will assume that when there are N s isotopes at time s, the solution is
You can check that we really do have the solution: when t = s the formula in (1.2) gives N( s) = N s, while we have
and so the differential equation (1.1) is satisfied.
It follows from (1.2) that the number of radioactive isotopes decays exponentially...
