Differential Equations
Michaelmas Term 2004
Stuart B Dalziel
Department of Applied Mathematics and Theoretical
Centre for Mathematical
Wilberforce Road
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
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.
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