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