Introduction

In physics, chemistry and the other sciences, we constantly build mathematical models of situations where a rate of change is known — how fast a population grows, how fast a radioactive substance decays, how fast a body cools — and the real question is to recover the underlying quantity itself as a function of time (or of whatever the independent variable is). Because the given information is a statement about a RATE, the model naturally comes out as an equation containing a derivative rather than the function directly. Finding a function that satisfies such an equation — i.e. finding the function whose derivative(s) match the given relation — is exactly the problem this chapter is built around. This short introduction motivates everything that follows: the definition of a differential equation, how to classify one by order and degree, how such equations arise in the first place from a family of curves or a physical law, the standard techniques for solving the common types (variables separable, homogeneous, and linear), and finally a tour of the real situations — population growth, radioactive decay, Newton's law of cooling, and rate-of-change geometry problems — where these equations are the natural modelling tool.