Ordinary Differential Equations

• Credit value: 30 credits at Level 6
• Convenor and tutor: Andrew Bowler
• Assessment: short, problem-based assignments (20%) and a three-hour examination (80%)

In this module we will study ordinary differential equations, which are used in mathematical modelling. We will develop solution methods and techniques, and explore some of the real-world applications.

Indicative module syllabus

• ODEs of order one: separation of variables, homogeneous differential equations, exacts differential equations, integrating factors, linear differential equations, change of variable
• ODEs of order two: linear differential equations with constant coefficient, homogeneous part, particular solutions, some families of linear second order differential equations, Euler equations
• Series and numerical solution: using power series and Taylor series to solve differential equations, Euler’s method, other single-step numerical methods
• The Laplace transform: definition and properties, using the Laplace transform to solve differential equations
• Dynamical systems: coupled differential equations, fixed points, phase portraits, linear systems, nonlinear systems, linearization, periodic orbits, limit cycles, chaotic systems
• Applications: exponential growth, population models, logistic growth, simple harmonic motion, damped and forced simple harmonic motion, predator-prey models, competing species, epidemics, mixing problems, heat loss

By the end of this module, you will have:

• knowledge and understanding of, and the ability to use, mathematical and/or statistical techniques
• knowledge and understanding of a range of results in mathematics
• an appreciation of the need for proof in mathematics, and the ability to follow and construct mathematical arguments
• an awareness of the use of mathematics and/or statistics to model problems in the natural and social sciences, and the ability to formulate such problems using appropriate notation
• an understanding of the importance of assumptions and an awareness of where they are used and the possible consequences of their violation
• an ability to present, analyse and interpret data
• knowledge and understanding of the processes and limitations of mathematical approximation and computational mathematics
• knowledge and understanding of a range of modelling techniques, their conditions and limitations, and the need to validate and revise models
• a deeper knowledge of some particular areas of mathematics, in particular, linear programming, game theory, difference equations, graphs and networks.