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

**Syllabus:**

Definitions. First-order differential equations:

linear, separable.

Second-order differential equations.

reduction of order, constant coefficients;

second-order linear equations: ordinary points and regular singular points.

Euler's equation.

Series solutions of second-order linear differential equations.

Power series, solutions about an ordinary point.

Solutions about a regular singular point.

Equal roots of indicial equation and roots differing by an integer.

Introduction to systems of first-order equations.

Two linear first-order equations.

Non-linear differential equations and stability.

Autonomous systems: trajectories in the phase plane, critical points.

Stability and asymptotic stability.

Linear and almost linear systems; classification of critical points.

Competing species and predator-prey problems.

**On completion of the course students should be able to:**

- use some of the standard methods for solution of first- and second-order ordinary differential equations;

- be aware of the implications of existence and uniqueness theorems;

- solve systems of linear first-order equations in two unknowns with constant coefficients;

- analyse the stability characteristics of non-linear systems in two unknowns.

- Module Supervisor: Georgi Grahovski
- Module Supervisor: Hadi Susanto