Module Content

Dimensional Analysis 

Units vs dimensions, the seven SI dimensions, dimensionless quantities, dimensional homogeneity. 

Difference Equations 

Classification of difference equations. 

First-order equations: methods of solution, equilibrium, stability, periodicity, cobweb diagrams. Applications in medicine: drug elimination, calculation of repeated dosages. Applications in finance: simple vs compound interest, amortization of loans, the cobweb theorem of economics. 

Second-order equations: linear equations with constant coefficients, Fibonacci’s rabbits. 

Differential Equations 

Classification of differential equations. 

First-order equations: integrating factors for linear equations, separable equations, homogeneous equations; using initial conditions. Modelling with first-order equations: population dynamics. 

Second-order equations: homogeneous equations with constant coefficients, distinct roots, repeated roots, complex roots; inhomogeneous equations: method of undetermined coefficients. 

Mechanics 

Physical concepts: force, work, energy, and momentum. Newton’s laws of motion; the second law as a second-order differential equation; derivation of the SUVAT equations. Kinematics in two dimensions. Simple harmonic motion, oscillations, resonance, and damping. 

Teaching Methods

The module will be delivered on campus, with weekly lecture and tutorial sessions. 

Printed notes will be given ahead of time for each section of the course, to support and enhance students’ preparation and engagement during class sessions. Lectures will follow the notes, with discussions of the main theoretical topics, and study of examples of the applications of the theory. There will be a strong emphasis on student involvement in discussions in lectures, to encourage a more active approach to learning the material, and to allow the delivery to be tailored to build on the students’ current understanding. 

Regular formative work in tutorial sessions will allow students to internalise the mathematical ideas and methods developed in the lectures, and lead to the development of problem-solving skills. This formative work will also feed back into the delivery of lectures and tutorials. 

Assessment Methods

The module is assessed through a portfolio of exercises and an examination.