Ordinary Differential Equations

The details
Mathematical Sciences
Colchester Campus
Undergraduate: Level 5
Monday 13 January 2020
Friday 20 March 2020
17 December 2019


Requisites for this module


MA225, MA323

Key module for

BSC N233 Actuarial Science (Including Placement Year),
BSC N323 Actuarial Science,
BSC N324 Actuarial Science (Including Year Abroad),
BSC N325 Actuarial Science (Including Foundation Year),
BSC L1G2 Economics and Mathematics (Including Placement Year),
BSC LG11 Economics and Mathematics,
BSC LG18 Economics and Mathematics (Including Foundation Year),
BSC LG1C Economics and Mathematics (Including Year Abroad),
BSC L1G1 Economics with Mathematics,
BSC L1G3 Economics with Mathematics (Including Placement Year),
BSC L1G8 Economics with Mathematics (Including Foundation Year),
BSC L1GC Economics with Mathematics (Including Year Abroad),
BSC GN13 Finance and Mathematics,
BSC GN15 Finance and Mathematics (Including Placement Year),
BSC GN18 Finance and Mathematics (Including Foundation Year),
BSC GN1H Finance and Mathematics (Including Year Abroad),
BSC G100 Mathematics,
BSC G102 Mathematics (Including Year Abroad),
BSC G103 Mathematics (Including Placement Year),
BSC G104 Mathematics (Including Foundation Year),
BSC 5B43 Statistics (Including Year Abroad),
BSC 9K12 Statistics,
BSC 9K13 Statistics (Including Placement Year),
BSC 9K18 Statistics (Including Foundation Year),
BSC G1G4 Mathematics with Computing (Including Year Abroad),
BSC G1G8 Mathematics with Computing (Including Foundation Year),
BSC G1GK Mathematics with Computing,
BSC G1IK Mathematics with Computing (Including Placement Year),
BSC G1F3 Mathematics with Physics,
BSC G1F4 Mathematics with Physics (Including Placement Year),
BSC GCF3 Mathematics with Physics (Including Year Abroad)

Module description

The subject of ordinary differential equations is a very important branch of Applied Mathematics. Many phenomena from Physics, Biology, Engineering, Chemistry, Finance, among others, may be described using ordinary differential equations. To understand the underlying processes, we have to find and interpret the solutions to these equations. The last part of the module is devoted to the study of nonlinear differential equations and stability.

The course provides an overview of standard methods for solving single ordinary differential equations and systems of ordinary differential equations, with an introduction to the underlying theory.

Module aims

The aim of the module is to introduce the students to the basic theory of ordinary differential equations and give a competence in solving ordinary differential equations by using analytical methods.

Module learning outcomes

On completion of the module students should be able to:

1. use some of the standard methods to solve first order ODEs;
2. use some of the standard methods to solve second order ODEs;
3. solve higher order linear ODEs;
4. solve systems of first order linear ODEs;
5. be aware of the implications of existence and uniqueness theorems.

Module information


1. Introduction, Some Basic Mathematical Models, Solutions of Some Differential Equations, Classification of Differential Equations, First order differential equations: Linear Equations with Variable Coefficients, Separable Equations.

2. First order differential equations: Differences Between Linear and Nonlinear Equations, Exact Equations and Integrating Factors, The Existence and Uniqueness Theorem.

3. Second Order Linear Equations: Homogeneous Equations with Constant Coefficients, Fundamental Solutions of Linear Homogeneous Equations, Linear Independence and the Wronskian, Complex Roots of the Characteristic Equation, Repeated Roots; Reduction of Order.

4. Second Order Linear Equations: Non-homogeneous Equations, Method of Undetermined Coefficients, Variation of Parameters.

5. Higher Order Linear Equations: General Theory of nth Order Linear Equations, Homogeneous Equations with Constant Coefficients, The Method of Undetermined Coefficients, The Method of Variation of Parameters.

6. Systems of First Order Linear Equations: Basic Theory of Systems of First Order Linear Equations, Homogeneous Linear Systems with Constant Coefficients, Complex Eigenvalues, Fundamental Matrices, Repeated Eigenvalues, Non-homogeneous Linear Systems.

7. Nonlinear Differential Equations and Stability: The Phase Plane, Linear Systems, Autonomous Systems and Stability

'A' level Maths or equivalent normally required.

Learning and teaching methods

This module runs at 3 contact hours per week. In the Summer term, 3 revision lectures are given.


  • William E. Boyce; Richard C. DiPrima; Douglas B. Meade. (2017) Boyce's elementary differential equations and boundary value problems, Hoboken, NJ: Wiley.

The above list is indicative of the essential reading for the course. The library makes provision for all reading list items, with digital provision where possible, and these resources are shared between students. Further reading can be obtained from this module's reading list.

Assessment items, weightings and deadlines

Coursework / exam Description Deadline Weighting
Written Exam  Test 1     
Written Exam  Test 2     
Exam  120 minutes during Summer (Main Period) (Main) 

Overall assessment

Coursework Exam
20% 80%


Coursework Exam
0% 100%
Module supervisor and teaching staff
Dr Murat Akman, email:
Dr Murat Akman, email
Dr Murat Akman (



External examiner

Dr Tania Clare Dunning
The University of Kent
Reader in Applied Mathematics
Available via Moodle
Of 70 hours, 66 (94.3%) hours available to students:
4 hours not recorded due to service coverage or fault;
0 hours not recorded due to opt-out by lecturer(s).


Further information
Mathematical Sciences

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