•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) Analyse existence and uniqueness of first-order initial-value problems. Solve the underlying ordinary differential equations (ODEs). Obtain solutions to second-order ODEs whose first solution is known and second-order ODEs whose coefficients are constant.

(LO2) Calculate eigenvalues and associated eigenfunctions of second-order homogeneous boundary-value problems (BVPs). Convert the underlying ODEs to Sturm-Liouville form to establish the orthogonality of their eigenfunctions. Solve second-order homogeneous ODEs with variable coefficients by the method of power series.

(LO3) Solve homogeneous systems of first-order linear ODEs, obtain their particular solutions when they are inhomogeneous by the methods of undetermined coefficients and diagonalisation.

(LO4) Solve inhomogeneous systems of first-order linear ODEs by the method quasi-diagonalisation in the case of incomplete sets of eigenvectors. Obtain solutions of three second-order partial differential equations (PDEs), namely the wave, heat and Laplace's equations, by the method of separation of variables.

(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.