Mathematics IA:
Differential Equations

Michaelmas Term 2004

Stuart B Dalziel

Department of Applied Mathematics and Theoretical Physics
University
of Cambridge
Centre for Mathematical Sciences
Wilberforce Road

Cambridge CB3 0WA

s.dalziel@damtp.cam.ac.uk

Examples sheets

These are on the DAMTP web site (http://www.damtp.cam.ac.uk/user/examples/).

Lecture notes

It is expected that students will attend lectures and take their own notes. While lecture notes are available for reference below, what is said in lectures will often go further and/or add more insight than is explicitly stated in the often terse description found in the lecture notes. Moreover, the lecture notes undoubtedly contain typos and misrepresentations that will hopefully be corrected in lectures.

Note: because html is not very good for mathematical typesetting, material will typically be provided as pdf files.

Lecture schedule

The first year course is intended to be accessible to able students who have a level of preparation equivalent to a single A-level in Mathematics, though a great majority of the students do have A-level Further Mathematics (or an equivalent qualification). It is not assumed that students have taken courses in physics, statistics or computing. However, applied mathematics and theoretical physics are included in the course, so Physics A-level is useful.

Introduction

Basic calculus

Differentiation as a limit, the chain rule, Leibnitz’ rule, elementary treatment of Taylor series; integration as an area, integration by substitution and parts; fundamental theorem of calculus; differentiation under integrals. [3]

First-order equations

Linear equations with constant coefficients: exponential growth, comparison with discrete equations, series solution; modelling examples including radioactive decay and time delay equation. Linear equations with non-constant coefficients: solution by integrating factor, series solution, comparison with discrete equations. [3]

Nonlinear equations: separable equations, families of solutions, isoclines, the idea of a flow and connection with vector fields, equilibrium solutions, stability by perturbation and phase-plane analysis; examples, including logistic equation and chemical kinetics; comparison with discrete equations including the logistic equation. [5]

Higher-order equations

Linear equations: complementary function and particular integral, linear independence, Wronskian, equations with constant coefficients and examples including radioactive sequences, comparison in simple cases with difference equations, reduction of order, resonance, transients, damping, response to step and impulse function inputs, coupled first order systems, series solutions including statement only of the need for the logarithmic solution. [8]

Nonlinear equations

Elementary phase-plane analysis, equilibrium and stability, examples including predator-prey systems. [5]

Appropriate books

W.E. Boyce and R.C. DiPrima Elementary Differential Equations and Boundary-Value Problems. Wiley 7th edition 2001 (£34.95 hardback). 8th ed. due for publication in May 2004

D.N. Burghes and M.S. Borrie Modelling with Differential Equations. Ellis Horwood 1981 (out of print).

W. Cox Ordinary Differential Equations. Butterworth-Heinemann 1996 (£14.99 paperback).

F. Diacu An introduction to Differential Equations: Order and Chaos. Freeman 2000 (£38.99 hardback).

N. Finizio and G. Ladas Ordinary Differential Equations with Modern Applications. Wadsworth 1989 (out of print).

D. Lomen and D. Lovelock Differential Equations: Graphics-Models-Data. Wiley 1999 (£80.95 hardback).

R.E. O’Malley Thinking about Ordinary Differential Equations. Cambridge University Press 1997 (£19.95 paperback).

D.G. Zill and M.R. Cullen Differential Equations with Boundary Value Problems. Brooks/Cole 2001 (£37.00 hardback). 11

 

Last updated: 11 November 2004