Module Content

Recap
First order separable and linear equations (integrating factor method), 2nd order homogeneous and inhomogeneous ODEs with constant coefficients.
Second Order Differential Equations
Wronskian, Abel’s Identity, general solution. Method of reduction of order. Inhomogeneous second order linear differential equations, variation of parameters. Euler-Cauchy ODEs.
Series Solutions of ODEs
Series solution of ordinary differential equations, use of Taylor Series and method of Frobenius.
Sturm-Liouville Theory
Adjoint, self-adjoint differential equations, normal version of the self-adjoint form. The Sturm-Liouville problem, eigenvalues and eigenfunctions, orthogonality of the eigenfunctions. Orthonormal systems, expansion of a function as a series of orthonormal eigenfunctions.
Systems of ordinary differential equations
Solutions of systems of linear equations with constant coefficients using eigenvalues and eigenvectors. Non-linear systems, phase diagrams, equilibrium points, stability. Linear approximation around equilibrium points.
Fourier Series
Odd, even & periodic functions. Fourier coefficients, Convergence of Fourier series, Dirichlet’s theorem, Summation of a series using Fourier series, Fourier series for even & odd functions, Fourier sine and cosine series, Fourier series of period 2l.
Solving ODEs with a computer algebra package
Solving ODEs analytically & numerically, series solutions, solving systems of ODEs, plotting solutions.

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.