Module Directory

This module 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. The first part is devoted to basic theory and methods for solving scalar ODEs. The second part of the module is devoted to the study of Systems of linear ODEs.

The subject of ordinary differential equations is a very important branch of Mathematical Analysis and has deep conections with 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 aims of this module are:

• To introduce students to the basic theory of ordinary differential equations.


• To give a competence in solving ordinary differential equations by using analytical methods.

By the end of the module, students will be expected 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. Be familiar with the basic theory and be able to solve higher order linear ODEs;


4. Be familiar with the basic theory and be able to solve systems of first order linear ODEs;


5. Be aware of the implications of existence and uniqueness theorems.

Indicative syllabus:

1. Revision of known knowledge on ODEs: 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, (Euler) homogeneous equations, Bernoulli equation, Riccati equation. Equilibrium solutions of autonomous ODEs. 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; The Existence and Uniqueness Theorem.


4. Second Order Linear Equations: ODEs with missing variables, 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 including Linear Systems with Constant Coefficients. The fundamental matrix of a system. The fundamental matrix as matrix exponential.

Teaching in the School will be delivered using a range of face-to-face lectures, classes, and lab sessions as appropriate for each module. Modules may also include online only sessions where it is advantageous, for example for pedagogical reasons, to do so.