**Content**: How do you reconstruct a curve given its slope at every point? Can you predict the trajectory of a tennis ball? The basic theory of ordinary differential equations (ODEs) as covered in this module is the cornerstone of all applied mathematics. Indeed, modern applied mathematics essentially began when Newton developed the calculus in order to solve (and to state precisely) the differential equations that followed from his laws of motion.

However, this theory is not only of interest to the applied mathematician: indeed, it is an integral part of any rigorous mathematical training, and is developed here in a systematic way. Just as a `pure' subject like group theory can be part of the daily armoury of the `applied' mathematician , so ideas from the theory of ODEs prove invaluable in various branches of pure mathematics, such as geometry and topology.

In this module we will cover relatively simple examples, first order equations, linear second order equations and coupled first order linear systems with constant coefficients, for most of which we can find an explicit solution. However, even when we can write the solution down it is important to understand what the solution means, i.e. its `qualitative' properties. This approach is invaluable for equations for which we cannot find an explicit solution.

We also show how the techniques we learned for second order differential equations have natural analogues that can be used to solve difference equations.

The course looks at solutions to differential equations in the cases where we are concerned with one- and two-dimensional systems, where the increase in complexity will be followed during the lectures. At the end of the module, in preparation for more advanced modules in this subject, we will discuss why in three-dimensions we see new phenomena, and have a first glimpse of chaotic solutions.

**Aims**: To introduce simple differential and difference equations and methods for their solution, to illustrate the importance of a qualitative understanding of these solutions and to understand the techniques of phase-plane analysis.

**Objectives**: You should be able to solve various simple differential equations (first order, linear second order and coupled systems of first order equations) and to interpret their qualitative behaviour; and to do the same for simple difference equations.

**Books**:

The primary text will be:

J. C. Robinson *An Introduction to Ordinary Differential Equations*, Cambridge University Press 2003.

Additional references are:

W. Boyce and R. Di Prima, *Elementary Differential Equations and Boundary Value Problems*, Wiley 1997.

C. H. Edwards and D. E. Penney, *Differential Equations and Boundary Value Problems*, Prentice Hall 2000.

K. R. Nagle, E. Saff, and D. A. Snider, *Fundamentals of Differential Equations and Boundary Value Problems*, Addison Wesley 1999.

This module aims to indicate to students how intuitive geometric and physical concepts such as length, area, volume, curvature, mass, circulation and flux can be translated into mathematical formulas. It also aims to teach the practical calculation of these formulas and their application to elementary problems in particle and fluid dynamics.