Differential Equations

Abstract

When, for example, we seek to model changes in population size, or the path of a projectile, or the escape of water down a plughole, or rowing across a river, we generally have information about the rate of change of one variable, y, with respect to another one, x; writing down that information often leads to an equation involving quantities such as $\frac{dy}{dx}$ or $\frac{d^2y}{dx^2}$, the derivatives of y with respect to x. In this introduction to a vast topic, we consider only straightforward first or second order ordinary differential equations: we show how they can be set up from verbal information, and how particular types can be solved by standard methods. We look at linked systems, with applications to predator-prey equations, and models for the spread of epidemics or rumours, with Exercises on topics such as carbon dating, cooling of objects, evaporation of mothballs, mixing of liquids and Lanchester’s Square Law about conflicts.