1. to provide the basic methods for modelling mathematically topics in subjects like biology, engineering, physical sciences and social sciences;

2. to discuss the advantages of using mathematics in modelling;

3. to demonstrate some simple models involving differential equations and difference equations;

After completing the module students should be able to:

. solve simple differential equations;

·    understand some methods of mathematical modelling and, in particular, the need to attach meaning to mathematical results;

·    develop some differential equations for population growth, and interpret the results;

·    understand Newton''s laws of Mechanics;

·    do simple problems in projectiles and orbits, some involving polar co-ordinates.

Philosophy of Applied Mathematics and of mathematical modelling.

Simple models of population growth. Difference equations. Solution and interpretation of the model of population growth.

The notion of a differential equation. First order ordinary differential equations with separable variables.

Simple queuing models.

Some simple two-variable models. Phase trajectories.

Linear and non-linear differential equations. Qualitative analysis of equations.

Foundations of mechanics. Simple models, including projectiles.

Friction. Simple proble ms involving air resistance.

Simple and damped harmonic oscillators. Second order linear differential equations with constant coefficients and simple free terms.

Description of orbital problems.