MC125 Ordinary Differential Equations

Explanation of Pre-requisites

Course Description

The `methods' part of this course may have some overlap with some A-level Further Maths syllabuses.

Aims

1.

To give students an appreciation of the fact that there is sophisticated mathematics behind the study of differential equations;

2.

To equip students with the basic skills required to deal with differential equations which can be solved in closed form by standard techniques, thus preparing them for later modules in their degrees.

Objectives

1.

Students should know how to define an initial value problem for a first order equation, and give examples of initial value problems which do not have unique solutions.

2.

Students should know basic techniques for solving first order ODEs of separable and linear types.

3.

Students should know how to solve second-order equations with constant coefficients.

4.

Students should be able to identify conservative second-order equations and use energy methods to analyze them.

Transferable Skills

Syllabus

1. First Order Scalar Equations

Definition of first order ODE y' = f(x,y) and of solutions of first order ODEs. Examples. Statement of Peano's Theorem (no proof). [1 lecture]

Solution techniques for simple first order ODEs, covering

• separable equations;
• linear equations (including integrating factors);
• nonlinear equations of the form y' = f(y/x) (if time permits);
• Bernoulli equations (if time permits).

Remark: We shall not cover so-called EXACT equations as these require a knowledge of partial derivatives. Also, exact equations usually only arise in mathematics exams. [4 lectures]

The need for initial conditions: the concept of an *initial value problem*. Examples. Statement of Picard's Theorem (no proof). Examples of failure of uniqueness in absence of Lipschitz continuity.
[2 lectures]

2. Second Order Scalar Equations

Definition of second order ODE y'' = f(x,y,y'). Examples. [1 lecture]

Second order linear equations with constant coefficients.

• Homogeneous equations. How to solve a homogeneous second order equation with constant coefficients by reference to the roots of the auxiliary quadratic. What to do in the case of coincident roots. What to do in the case of complex roots, including the case of purely imaginary roots. (Note: this can be done without complex exponentials.) Examples, including damped oscillations of a spring.[4 lectures]
• Inhomogeneous equations. How to find particular integrals for various standard forms of inhomogeneity. The variation-of-parameters method. Examples, including forced damped oscillation of a spring. [3 lectures]

Second order conservative equations. Energy methods for reduction to a separable first order equation. Examples, including bungee jumping without injury. [3 lectures]

TOTAL: 18 lectures.

Reading list

Background:

R. Bronson, Schaum's Outline of Theory and Problems of Differential Equations, McGraw-Hill, 1994.

F. R. Giordano and M. R. Weir, Differential Equations, A Modelling Approach Addison Wesley, 1991.

A. Jeffrey, Linear Algebra and Ordinary Differential Equations

D. Pearson, Calculus and ODE's, Arnold, 1996.

J. Gilbert, Guide to Mathematical Methods, Macmillan, 1991.

Details of Assessment