# BSc Mathematics with Placement Year

Year of entry: 2024

## Course unit details:ODEs and Applications

Unit code MATH11422 20 Level 1 Semester 2 No

### Overview

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.

### Aims

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. Define the main physical quantities of classical mechanics (force, velocity, acceleration, momentum etc)
5. Discuss Newton's laws of motion and gravity.
6. Introduce simple conservative and non-conservative systems involving single particles.
7. Introduce model problems leading to ODEs.

### Learning outcomes

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 determine their stability. Perform phase plane analysis.
• Derive equations of motion for a single particle in rectilinear and circular motion. Derive integrals of motion and relate them to conserved quantities.
• Analyse the damped, driven harmonic oscillator using second order ODEs with constant coefficients.
• Analyse and interpret population models described by first order systems of ODEs.
• Apply and interpret the results of a simple numerical method (the Euler method) for approximately solving initial value problems.

Method Weight
Other 30%
Written exam 70%

### Feedback methods

There is a supervisions each week which provides an opportunity for students; work to be marked and discussed and 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 The Applicaations
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/
R.D. Gregory — Classical Mechanics (CUP)

### Study hours

Scheduled activity hours
Lectures 36
Tutorials 12
Independent study hours
Independent study 152

### Teaching staff

Staff member Role
Marcus Webb Unit coordinator
Gareth Wyn Jones Unit coordinator