Building upon a general understanding of the form and usefulness of ordinary differential equations and knowledge of elementary solution methods, this module explores the mathematical foundations of ordinary differential equation theory as well as methods for the asymptotic and qualitative study of their solutions.

It is an intriguing observation that only a very small number of types of differential equation can be solved in terms of the well-known elementary functions. Differential equations are therefore a fruitful source of new functions and thus are of great practical value in applications and remain of continuing interest. However, this also means that mere knowledge of techniques for the explicit solution of differential equations will not reach very far.

It is therefore essential to have a theoretical framework which ensures the existence of solutions of ordinary differential equations without the need to find them explicitly, and to study the uniqueness and continuous dependence of solutions on parameters of the equation. In the presence of singularities, the asymptotic behaviour of solutions is very valuable information. A further aspect of the qualitative study of ordinary differential equations is the question of stability: will nearby starting points lead to wildly different solutions (chaos), or will the solutions approach a fixed point or an attractive set of more complicated structure, e.g. a limit cycle?

The module will provide an introduction to the existence theory of ordinary differential equations and to fundamental techniques of the asymptotic and qualitative study of their solutions.

• Appreciate the limitations of explicit solution methods in ODE analysis.
• Know and understand the existence, uniqueness and continuity theory for ODEs, including proofs.
• Understand and perform asymptotic analysis of ODE solutions.
• Know, understand and apply techniques of stability analysis to ODEs.

27 - 50 minute lectures

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 50 hours private study including preparation of solutions to given exercises.

Skills:

Correct manipulation of analytical problems of differential equation type.

Knowing the relative merits of explicit vs. qualitative/asymptotic methods and ability to apply each as appropriate.

Understanding the form, value and necessity of mathematical proof.

Ability to construct correct logical arguments.

Ability to analyse ODE problems using appropriate techniques.

Vision of scope of ODE models. 

Transferable Skills:

Understanding the value of qualitative (as opposed to explicit) analysis.

Practising correct formal manipulation and logical thought.

Training mental visualisation.

Preciseness and clarity of expression.

Ability to recognise, formulate and solve problems in an interdisciplinary environment.

Formative assessment is carried out by means of regular homeworks. Feedback to students on their progress towards learning outcomes is provided by checking homeworks and during exercise lectures.

The summative assessment is the written examination at the end of the module. This gives students the opportunity to demonstrate their overall achievement of learning outcomes. It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has two sections of equal weight. Section A contains a number of compulsory questions of variable length but normally short. Section B has a choice of two from three equally weighted questions.

• Elementary solution methods for ODE
• The Picard-Lindeloef existence and uniqueness theorem, continuous dependence on
  parameters, Peano's existence theorem and counterexamples to uniqueness
• Linear systems, Floquet theory of periodic systems
• Asymptotic stability, Lyapunov functions, limit cycles

Please see Background Reading List for an indicative list.

Differential Equations. Introduction and Qualitative Theory, Cronin, J., Marcel Dekker, New York 1980

Differential Equations, Dynamical Systems, and Linear Algebra, Hirsch,M. W., Smale, S., Academic Press, San Diego,1974

Differential Equations with Applications and Historical Notes, Simmons, G. F., McGraw-Hill, New York, 1991

Ordinary Differential Equations, Hartman, P., Wiley, New York, 1964