[The University of Leicester]

Department of Mathematics



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MA3061 Ordinary Differential Equations

MA3061 Ordinary Differential Equations

Credits: 20 Convenor: Dr. M. Tretyakov Semester: 1 (weeks 1 to 12)
Prerequisites: essential: MA1002(=MC127), MA1151(=MC146), MA1152, MA2101, MA2102 desirable: MA2121
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

Subject Knowledge

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.

Learning Outcomes

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

Methods

Class sessions together with some handouts.

Assessment

The final assessment of this module will consist of 10% coursework and 90% from a three hour examination. 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.

Subject Skills

Aims

Develop problem solving skills, written communication skills, modelling skills.

Learning Outcomes

Students will have to investigate problems, draw conclusions and make conjectures. Students will be able to use the techniques taught within the module to solve problems, and be able to present arguments and solutions in a coherent and logical form. 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.

Methods

Class sessions, coursework, exam.

Assessment

Marked problem sheets, exam.

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.

Syllabus

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

Reading list

Recommended:

E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill. D. W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations, Oxford University Press. F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer. V.I. Arnold, Ordinary Differential Equations, Springer. D.K. Arrowsmith and C.M. Place, Dynamical Systems, Chapman&Hall. P. Hartmann, Ordinary Differential Equations, Wiley. W.E. Boyce and R.C. Di Prima, Elementary Differential Equations and Boundary Value Problems, Wiley.

Resources

Problem sheets, additional handouts, lecture rooms.

Module Evaluation

Module questionnaires, module review, year review.

Next: MA3101 Abstract Algebra Up: ModuleGuide03-04 Previous: MA3011 Applied Numerical Mathematics

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Last updated: 2004-02-21
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