Content:

1. Introduction to mathematical modelling with differential equations: Modelling cycle, principles and observations, types of problems, scaling and dimensional analysis, simplification and reduction, perturbation methods.
2. Intro to differential equations: Classification, general first order equations, autonomous equations, stability, integrating factors for linear equations, separation and substitution methods for nonlinear equations.
3. Systems and higher order equations: Relation between higher order equations as systems, general 2x2 systems, autonomous systems, phase portraits, linearisation and linear stability, general theory for linear systems, eigenspace analysis in case of constant coefficients.
4. Further problems and techniques: a selection from discretisation principles and difference equations, control problems, dynamical systems, attractors and linearisation.

Learning Outcomes: By the end of the module students should be able:

• To understand the modelling cycle in science and engineering, to formulate mathematical models and problems using differential equations, and to use a variety of methods to reveal their main underlying dynamics.
• To apply a range of techniques to solve simple ordinary differential equations (first order, second order, first order systems), and to gain insight into the qualitative behaviour of solutions.
• To confidently deploy computational methods and software to validate results, to approximate solutions of more challenging problems, and to further investigate them.

Books:

Robinson, James C. An Introduction to Ordinary Differential Equations. Cambridge University Press, 2004.

Witelski, B. and Bowen, M., Methods of Mathematical Modelling: Continuous Systems and Differential Equations. Springer, 2015.

Logan, David. A First Course in Differential Equations. Springer, 2015.

Holmes, Mark H. Introduction to the Foundations of Applied Mathematics. Springer, 2019.