MMath&Phys Mathematics and Physics / Course details

The unit provides a basic introduction to ordinary differential equations (ODEs) and some applications. The course will discuss both methods, including analytical as well graphical and approximate methods associated with general first and second order ODEs. Applications including simple problems drawn from many fields including Newtonian mechanics, population models, economics, biology, will be introduced, and the corresponding equations will be written down and their solutions discussed.

The unit provides a basic introduction to ordinary differential equations (ODEs) and applications. The course will discuss both methods, including analytical as well graphical and approximate methods associated with general first and second order ODEs. Applications including simple problems drawn from many fields including Newtonian mechanics, population models, economics, biology, will be introduced, and the corresponding equations will be written down and their solutions discussed.

The main aims are to:

1. Provide a classification of ODEs
2. Provide analytical methods of solving both first and second-order ODEs;
3. Introduce approximate methods (graphical, numerical, approximate) for solving first order and systems of first order equations.
4. Introduce model problems leading to ODEs.

On the successful completion of the course, students will be able to:

• Classify ODEs (in terms of order, linear/nonlinear autonomous/nonautonomous, initial or boundary value problem) and assess the existence and uniqueness of their solutions.
• Select and apply techniques for finding analytical solutions for several classes of first and second order ODEs.&
• Locate equilibrium points for first order systems of ODEs and perform phase plane analysis.&
• Apply and interpret the results of a simple numerical method (the Euler method) for approximately solving initial value problems.

There are supervisions each week for the first half of the semester which provide an opportunity for students; work to be marked and discussed and to provide feedback on their understanding. Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

Farlow - Introduction to Differential Equations and Their Applications  

Martin Braun (1993), Differential Equations and Their Applications, 4th edition, Volume 11 of Springer Texts in Applied Maths. DOI: 10.1007/978-1-4612-4360-1 

A good reference for extra reading is Trefethen, Birkisson, Driscoll – Exploring ODEs  (https://people.maths.ox.ac.uk/trefethen/ExplODE/)