Introduction

Many real situations — the motion of a projectile, rocket, satellite or planet; the current in an electric circuit; the conduction of heat along a rod; the vibration of a wire or membrane — force us to describe how a quantity's rate of change (its derivative) is related to the quantity itself and to other variables. Once such a rate law is written down mathematically, it becomes a differential equation: an equation involving one or more derivatives.

Some sample rate relationships between an unknown function $y$ and the independent variable $x$:

• Rate of change of $y$ directly proportional to $y$: $\dfrac{dy}{dx}=ky$.
• Rate of change of $y$ directly proportional to $y^2x$: $\dfrac{dy}{dx}=ky^2x$.
• Rate of change of $y$ inversely proportional to $y$: $\dfrac{dy}{dx}=\dfrac{k}{y}$.
• Rate of change of $y$ directly proportional to $y^2$ and inversely proportional to $x$: $\dfrac{dy}{dx}=\dfrac{ky^2}{x}$.

A differential equation is any equation in which some derivative(s) of an unknown function occur. In many physical models the independent variable is time.

To apply mathematics to a real problem we must first build a mathematical model — and because rates of change are represented by derivatives, that model very often turns out to be a differential equation relating an unknown function to one or more of its derivatives. Differential equations are therefore of basic importance across science and engineering: they describe population growth, radioactive decay, and a great deal of biology and economics besides.

The subject was invented, together with calculus itself, by Newton and Leibniz to solve problems in geometry and physics, and was developed further by the Bernoulli family, Euler, and others as part of Newtonian physics. Differential equations remain everywhere today — in mobile phones, motor cars, aircraft, weather forecasting, the internet, and healthcare.

This chapter introduces first-order ordinary differential equations, the standard methods (variables separable, substitution, integrating factor) used to solve them, and some of their real-life applications.