# Mathematics

## MTH2003 - Differential Equations (2019)

MODULE TITLE CREDIT VALUE Differential Equations 15 MTH2003 Prof Vadim N Biktashev (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11 weeks 0 0
 Number of Students Taking Module (anticipated) 295
DESCRIPTION - summary of the module content

Differential equations are at the heart of nearly all modern applications of mathematics to natural phenomena. Computerised applications play a vital role in many areas of modern technology. Mathematically, all rates of change and acceleration can be described by derivative functions. These include the growth of populations, the spread of diseases, movement of physical objects in response to forces acting on them, or even the fluctuations of the stock market. You will learn the basic principles of differential equations, and will apply that knowledge to some every day phenomena. Then you will learn about methods of finding solutions for some classes of differential equations.

This course will enable you to demonstrate an understanding of, and competence in, a range of analytical tools for posing and solving differential equations, and their application to engineering situations.

Prerequisite modules: MTH1002 or NSC1002 (Natural Science Students) or equivalent.

AIMS - intentions of the module

The aim of this module is to introduce you to some representative types of ordinary and partial differential equations and to introduce a number of analytical techniques used to solve them exactly or approximately.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:

1 demonstrate a working knowledge of how to identify, classify and solve a range of types of ordinary and partial differential equations;

2 reveal an insight into their application and derivation;

3 show some knowledge of a selection of special functions and series methods used for solution of these differential equations.

Discipline Specific Skills and Knowledge:

4 exhibit an understanding of range of analytical tools for posing and solving differential equations;

5 display competence in applying these tools;

6 prove an understanding of the concept of using differential equations for mathematical modelling of natural phenomena and engineering applicatoins.

Personal and Key Transferable/ Employment Skills and  Knowledge:

7 demonstrate an ability to monitor your own progress and to manage time;

8 show an ability to formulate and solve complex problems.

SYLLABUS PLAN - summary of the structure and academic content of the module

- Review of simple methods for solving first order ordinary differential equations (ODEs). Sufficient conditions to guarantee existence and uniqueness to such ODEs.

- The general linear second order ODE. Equations with constant coefficients, Euler-Cauchy equations. Reduction of order, variation of parameters.

- Power series methods: Leibniz-Maclauren method, method of Frobenius.

- Boundary value problems, eigenfunctions and eigenvalues. Orthogonality of eigenfunctions of Sturm-Liouville problems.

- Selected orthogonal systems of functions: trigonometric functions, Legendre polynomials, Bessel functions. Series in orthogonal functions, including Fourier series.

- Examples of linear partial differential equations (PDEs) and their solutions, including initial/boundary-value problems for one-dimensional wave equation and one-dimensional heat equation, on the whole real line and on a finite interval.

- Solution of PDEs in two and three spatial dimensions using separation of variables, a.k.a. normal modes or (generalised) Fourier series, in Cartesian, polar and spherical coordinates.

- Selected applications of ODEs and PDEs.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study 38 112
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled learning and teaching activities 33 Lectures including examples classes Scheduled learning and teaching activities 5 Tutorials Guided independent study 112 Lecture and assessment preparation; wider reading

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Exercise sheets 5 x 10 hours All Discussion at tutorials and solutions provided in ELE; tutor feedback on submitted solutions.

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams 0 100
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written exam – closed book 100 2 hours All Via SRS

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-reassessment
All above Written exam (100%) All August Ref/Def period

RE-ASSESSMENT NOTES

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

Web based and Electronic Resources:  ELE: http://vle.exeter.ac.uk

Reading list for this module:

Type Author Title Edition Publisher Year ISBN Search
Set Arfken G.B. & Weber H.J. Mathematical Methods for Physicists Electronic Harcourt/ Academic Press 2005 000-0-120-59825-6 [Library]
Set O'Neil P.V. Advanced Engineering Mathematics 2nd Wadsworth 1987 000-0-534-06792-1 [Library]
Set Stephenson G. & Radmore P.M. Advanced mathematical methods for engineering and science students Cambridge University Press 1990 000-0-521-36860-X [Library]
Set Boyce, W E, Di Prima, R C Elementary differential equations and boundary value problems 9th edition John Wiley and Sons 2009 978-0-470-39873-9 [Library]
Set kreyszig E Advanced Engineering Mathematics 9th Wiley 2006 978-0471728979 [Library]
CREDIT VALUE ECTS VALUE 15 7.5
PRE-REQUISITE MODULES MTH1002
NQF LEVEL (FHEQ) AVAILABLE AS DISTANCE LEARNING 5 No Tuesday 10 July 2018 Thursday 27 June 2019
KEY WORDS SEARCH Differential equations; orthogonal functions.