Introductory description

It is a far stream version of the new M^3-1 and M^3-2 (MA144, MA145, MA146,MA147) module that allows students access to differential equations (ODEs and PDEs) in year 2. It covers the differential equations topics covered in these modules as well as foundations of differentiable calculus.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

1. First order linear equations: first order linear equations, examples of existence and unique-ness, integration techniques (integrating factors, ...).
2. Second order equations: general homogeneous equations and linear second order equations with constant coefficients, reduction to 2x2 systems, sketching the flow under a vector field (2d only, phase diagrams).
3. Nonlinear equations and 2x2 systems: linear stability such as predator and prey models.
4. Discretisation techniques: explicit and implicit Euler, stability (optional).
5. Differential calculus of scalar functions of several variables: partial derivative, chain rule, change of coordinates, gradient, directional derivative.
6. Differential calculus of vector functions of several variables: crossed product, vector fields, divergence, curl, nabla, algebraic identities, Laplace, expression in other coordinates (polar, cylindrical, spherical), parametrisation of surfaces, tangent planes, normal
7. Integration: 2D-divergence theorem.

Learning outcomes

By the end of the module, students should be able to:

• You should be able to solve various simple differential equations (first order, linear second order and coupled systems of first order equations) and to interpret their qualitative behaviour;
• and to do the same for simple difference equations.

Indicative reading list

The primary text will be:
J. C. Robinson An Introduction to Ordinary Differential Equations, Cambridge University Press 2003.
Additional references are:
W. Boyce and R. Di Prima, Elementary Differential Equations and Boundary Value Problems, Wiley 1997.
C. H. Edwards and D. E. Penney, Differential Equations and Boundary Value Problems, Prentice Hall 2000.
K. R. Nagle, E. Saff, and D. A. Snider, Fundamentals of Differential Equations and Boundary Value Problems, Addison Wesley 1999.

Subject specific skills

See learning outcomes.

Transferable skills

Students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence.