# MA133 Differential Equations

**Lecturer: **Dwight Barkley

**Term:** Term 2

**Status for Mathematics students:** Core

**Commitment:** 30 lectures

**Assessment:** 15% from fortnightly assignments, 85% from a 2 hour examination

**Formal registration prerequisites: **None

**Assumed knowledge: **A-level mathematics or equivalent, in particular Calculus topics from Pure

**Useful background:** Proficiency with Mechanics from Maths A-level, or having taken Physics A-level, useful but not essential, we will cover necessary topics from first principles

**Synergies: **This module leads on to any module using differential or partial differential equations, most immediately MA134 Geometry and Motion in Term 2

**Leads to: **The following modules have this module listed as **assumed knowledge** or **useful background:**

- MA254 Theory of ODEs
- MA261 Differential Equations: Modelling and Numerics
- MA250 Introduction to Partial Differential Equations
- MA269 Asymptotics and Integral Transforms
- MA256 Introduction to Mathematical Biology
- MA258 Mathematical Analysis III
- MA209 Variational Principles
- MA390 Topics in Mathematical Biology
- MA3J3 Bifurcations, Catastrophes and Symmetry
- MA3H7 Control Theory
- MA4J1 Continuum Mechanics
- MA4M9 Mathematics of Neuronal Networks

**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.