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Section 3.3 Exponential Growth and Decay — a First Look at Differential Equations

A differential equation is an equation for an unknown function that involves the derivative of the unknown function. For example, Newton's law of cooling says:

The rate of change of temperature of an object is proportional to the difference in temperature between the object and its surroundings.

We can write this more mathematically using a differential equation — an equation for the unknown function \(T(t)\) that also involves its derivative \(\diff{T}{t}(t)\text{.}\) If we denote by \(T(t)\) the temperature of the object at time \(t\) and by \(A\) the temperature of its surroundings, Newton's law of cooling says that there is some constant of proportionality, \(K\text{,}\) such that

\begin{align*} \diff{T}{t}(t) &= K\big[T(t)-A\big] \end{align*}

Differential equations play a central role in modelling a huge number of different phenomena, including the motion of particles, electromagnetic radiation, financial options, ecosystem populations and nerve action potentials. Most universities offer half a dozen different undergraduate courses on various aspects of differential equations. We are barely going to scratch the surface of the subject. At this point we are going to restrict ourselves to a few very simple differential equations for which we can just guess the solution. In particular, we shall learn how to solve systems obeying Newton's law of cooling in Section 3.3.2, below. But first, here is another slightly simpler example.