## 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*; 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 they 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 of conflicts.

## References and Further Reading

- Daley D J and Kendall D G (1965) Stochastic Rumours.
*Journal of the Institute of Mathematics and its Applications*1 pages 42–55Google Scholar - Kermack W O and McKendrick A G (1927) A contribution to the Mathematical theory of Epidemics.
*Proceedings of the Royal Society A*115 pages 700–721Google Scholar - Lanchester F W (1916) Aircraft in warfare: the Dawn of the Fourth Arm. Constable and Co.Google Scholar
- Lotka A J (1910) Contribution to the Theory of Periodic Reactions.
*Journal of Physical Chemistry*14 pages 271–4Google Scholar - Simmons G F (1972) Differential Equations, with Applications and Historical Notes. McGraw-Hill.Google Scholar
- Volterra V (1926) Fluctuations in the abundance of species considered mathematically.
*Nature*118 pages 558–560Google Scholar