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.

Modules will be delivered through blended learning. You will be guided through learning activities appropriate to your module, which may include:

• Weekly face to face classes (e.g. labs, lectures, exercise classes)
• Electronic resources that you work through at your own pace (e.g. videos, exercise sheets, lecture notes, e-books, quizzes)

Students are also expected to undertake self-guided study throughout the duration of the module.

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.

• 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