•To familiarize students with basic ideas and fundamental techniques to solve ordinary differential equations.

•To illustrate the breadth of applications of ODEs and fundamental importance of related concepts.

(LO1) To understand the basic properties of ODE, including main features of initial value problems and boundary value problems, such as existence and uniqueness of solutions.

(LO2) To know the elementary techniques for the solution of ODEs.

(LO3) To understand the idea of reducing a complex ODE to a simpler one.

(LO4) To be able to solve linear ODE systems (homogeneous and non-homogeneous) with constant coefficients matrix.

(LO5) To understand a range of applications of ODE.

(S1) Problem solving skills

(S2) Numeracy

First-order ODEs: Theorem of existence & uniqueness (without proof). Separable, homogeneous, linear, exact equations.
Second-order ODEs: Theorem of existence & uniqueness (without proof). Homogeneous equations with special attention to linear equations, superposition principle, linear independence and the Wronskian determinant, Abel’s theorem, constant coefficients with a detailed example on a linear oscillator, reduction of order. Inhomogeneous equations with special attention to linear equations, method of undetermined coefficients with a detailed example on a forced linear oscillator, variation of parameters. (Euler equation to be covered in an Assignment.) Boundary-value problems and eigenfunctions, introduction to Sturm-Liouville (SL) Theory including orthogonality of SL eigenfunctions, eigenfunction expansions with an example on Fourier series. Power series solution at ordinary points. (Series solution at regular singular points may be covered in an Assignme nt.)
Systems of First-order ODEs: Theorem of existence & uniqueness (without proof). Homogeneous systems. Inhomogeneous systems covering method of undetermined coefficients, complete sets of eigenvectors (diagonalisation) and incomplete sets of eigenvectors (quasi-diagonalisation). (Optional Reading: Method of variation of parameters.)
First-order PDEs: Characteristics, quasi-linear equations.
Second-order PDEs: The wave, the heat and Laplace equations.