	Department of Mathematics & Computer Science

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MA3061 Ordinary Differential Equations

MA3061(=MC380) Ordinary Differential Equations

Credits: 20

Convenor: Dr. M. Tretyakov

Semester: 1

Prerequisites:	essential: MA1002(=MC127), MA1151(=MC146), MA2101(=MC248), MA2102(=MC241), MA2151(=MC240)	desirable: MC243
Assessment:	Coursework: 10%	Three hour exam: 90%

Lectures:	36	Problem Classes:	10
Tutorials:	none	Private Study:	104
Labs:	none	Seminars:	none
Project:	none	Other:	none
Surgeries:	none	Total:	150

Explanation of Pre-requisites

Students should already be familiar with the standard elementary methods for solving simple ODEs (MA1002). They will require a basic knowledge of real analysis (MA1151 and MA2101). For those parts of the course which deal with linear equations, students should be familiar with the idea of linear independence of elements of a vector space (MA2102) and with complex numbers. Finally it may be helpful for some parts of the course that students be familiar with norms of matrices (MA2121).

Course Description

The theory of ordinary differential equations (often given the designer label `dynamical systems' in recent years) is vast, and so the choice of topics for this course is very much determined by the lecturer's own interests.

The course will start with two basic topics which are required for almost everything else (existence and uniqueness theorems, and linear systems of equations). The remaining two topics (phase portraits and stability analysis of differential equations) will build on the knowledge acquired in the first two sections.

Aims

This course will consider, at an introductory level, four classical topics from the extensive theory of ODEs. By the end of the course the student should know enough to be able to engage in further independent study at postgraduate level in each of the areas covered.

Objectives

By the end of this module a diligent student should be able to

analyze initial value problems in order to determine whether or not they have unique solutions, and over what interval the existence of such solutions may be guaranteed;
know the basic properties of fundamental matrices for linear systems of differential equations, and be able to construct such matrices explicitly in simple cases, using them to find solutions of inhomogeneous systems;
use phase portraits to analyze problems which cannot be solved in closed form;
analyse stability of non-linear systems.

Transferable Skills

Most of the skills acquired on this module are transferable to other branches of pure and applied mathematics where differential equations arise; the course is a `desirable' prerequisite for our Level 4 course on Partial Differential Equations.

Syllabus

Existence and Uniqueness Theorems for solutions of initial value problems, including Picard's Theorem and variants thereof. Dependence of solutions on initial data.
Linear Systems of ordinary differential equations. Techniques for solving inhomogeneous equations.
Phase portraits of autonomous differential equations: analysis of critical points, and criteria for existence of periodic solutions and limit cycles.
Stability by linearization and by the direct method (Lyapunov functions).

Reading list

Details of Assessment

The final assessment of this module will consist of 10% coursework and 90% from a three hour examination during the Summer exam period. The 10% coursework contribution will be determined by students' solutions to coursework problems. The examination paper will contain 8 questions with full marks on the paper obtainable from 5 complete answers.