Section Summary

In this section we have studied a cooling body problem as an initial value problem. We have seen that application of mathematics to a real problem involves at least these three steps:

Step 1: Translate the problem into mathematical form.

Step 2: Solve the mathematical problem.

Step 3: Interpret the mathematical solution.

Our particular problem had two interesting features that will reappear from time to time.

First, we encountered decaying exponentials, functions of time `t` of the form $e^{-kt}$, where the constant `k` is positive. These functions have the property that the functional values approach zero as `t` becomes large. Such functions appear often in models of transient effects, that is, things that die out as time goes on.

Second, we made a change of the dependent variable — replacing `T` by $y=T-21$ — to simplify our problem to one we had seen before. Simplification to an already-solved problem is a standard problem-solving technique. It makes it unnecessary to keep doing the same work over and over.