Module code: MAT2049

Module provider


Module Leader

GOURLEY SA Prof (Maths)

Number of Credits


ECT Credits



FHEQ Level 5

JACs code


Module cap (Maximum number of students)


Module Availability

Semester 2

Overall student workload

Independent Study Hours: 117

Lecture Hours: 33

Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test IN-SEMESTER TEST (50 MINS) 20
Examination EXAMINATION 80

Alternative Assessment


Prerequisites / Co-requisites

MAT1036 Classical Dynamics; MAT2007 Ordinary Differential Equations

Module overview

To develop further the students knowledge of the applications of differential equations to problems arising in various contexts.

Module aims

develop further the students' understanding of the applications of differential equations to real world problems;

provide a broader knowledge of analytical techniques that are commonly used to tackle such equations.

Learning outcomes

Attributes Developed
An enhanced feel for how differential equations arise in modelling real world phenomena KCT
An appreciation of how to obtain quantitative information about the solutions of those equations KC
An appreciation of how to interpret the results ; KCT
An enhanced knowledge of analytical tools commonly used in the study of applied problems. KC

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

Module content

Applications of linear systems of the form dx/dt = Ax to chemical mixing problems; the idea of compartmental analysis. Engineering applications of second order systems of the form d2 x/dt2+A x=0, e.g. to railway carriages connected by springs.   Forced systems of the form d2 x/dt2+A x=f(t); resonance.
The Laplace transform and its properties. The Heaviside and Dirac delta functions and applied problems that give rise to differential equations containing those functions, for example the bending of a supported beam subject to a load at a particular point,  and mathematical models of drug therapy.
Calculus of Variations. The Euler-Lagrange equation. Examples to include the brachistochrone, the catenary (hanging cable) and isoperimetric problems. Variational problems with constraints: Lagrange multipliers and endpoint conditions.
Applications of nonlinear equations. Phase plane techniques applied to problems such as the nonlinear pendulum equation and van der Pol's equation. Simple analytical techniques for weakly nonlinear equations.

Methods of Teaching / Learning

The learning and teaching strategy is designed to provide:

 A detailed knowledge of analytical techniques relevant to the solution of differential equations arising in applied contexts;
Examples,  considered in detail, of the modelling of real life problems which can be described by those differential equations;
Solutions of those example problems and an appreciation of how we can interpret the results.

The learning and teaching methods include:

3 one hour lectures per week for 11 weeks, each session being a blend of traditional lecturing and class discussion.

Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate

·         Adequate knowledge of analytical techniques for solution of differential equations that arise in applied problems.

·         Skills in modelling real-life problems.

·         Skills in interpreting the results of the analysis.


Thus, the summative assessment for this module consists of:

·         One two-hour examination worth 80%.

·         One in-semester test, approximately in week 6, worth 20%.

Formative assessment and feedback

This takes the form of four marked exercise sheets issued at roughly equal intervals throughout the course. Written feedback is provided. 

Reading list


Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics with Statistics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics MMath 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MMath 2 Optional A weighted aggregate mark of 40% is required to pass the module

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2017/8 academic year.