Differential Equations - 2023 entry

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. In particular you will develop methods to solve the wave and heat equations which are fundamental to modelling many physical processes.

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 situations in science and technology. 

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

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.

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 applications.

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.

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

- Systems of first-order differential equations. Linear Systems. Constant coefficients case. Variation of Parameters.
 

- The general linear second order ODE. Equations with constant coefficients, Euler-Cauchy equations. Reduction of Order and Variation of Parameters.

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

- Power series methods: Leibniz-Maclauren method and method of Frobenius (selected examples).
 

- 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.

- 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.

Deferrals: Reassessment will be by coursework and/or exam in the deferred element only. For deferred candidates, the module mark will be uncapped

Referrals: Reassessment will be by a single written exam worth 100% of the module only. As it is a referral, the mark will be capped at 40%.   

Web based and Electronic Resources:  ELE: https://ele.exeter.ac.uk

Reading list for this module: