# 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: ideas from the theory of ODEs prove invaluable in various branches of pure mathematics, such as geometry and topology. The first half of this module will focus on ordinary differential equations - how to understand them and how to solve them. The second half of the module covers topics from multivariable calculus - partial derivatives, div, grad, curl, and some differential geometry and integration needed for subsequent modules on differential equations.

**Aims**: To introduce simple differential equations and methods for their solution and to provide a solid foundation in the calculus needed to study future modules involving ordinary and partial differential equations.

**Objectives**: You should be able to solve various simple differential equations (first order, linear second order and coupled systems of first order equations), be able to manipulate div, grad, and curl operations, and be able to integrate over simple curves and surfaces.

**Books**:

The primary text will be:

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

J. Stewart, D. Clegg, S. Watson, *Multivariable Calculus,* *ninth edition, metric version*, Brooks/Cole 2022.

Additional references are:

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