•To give an insight into some specific methods for solving important types of ordinary differential equations.

•To provide a basic understanding of the Calculus of Variations and to illustrate the techniques using simple examples in a variety of areas in mathematics and physics.

•To build on the students'' existing knowledge of partial differential equations of first and second order.

(LO1) Use the method of Variation of Arbitrary Parameters to find the solutions of some inhomogeneous ordinary differential equations

(LO2) Solve simple integral extremal problems with and without constraints

(LO3) Classify a system of simultaneous first-order linear partial differential equations, find the Riemann invariants and general or specific solutions in appropriate cases.

(LO4) Classify second-order linear partial differential equations and find general or specific solutions.

(LO5) Determine conformal mappings and use the conformal mapping for solving boundary value problems for the Laplace equation in complicated domains.

(LO6) Determine Fourier transforms of functions, apply Fourier transforms for solving boundary value problems for second order PDEs.

Ordinary differential equations; some methods, including the variation of arbitrary constants, for solving certain types of equations.

Introduction to the Calculus of Variations for problems without and with constraints.

Simultaneous first-order linear partial differential equations; Riemann invariants.

Second-order linear partial differential equations; classification, reduction to standard forms, conformal mappings. Fourier transforms.