Differential Equations module (MA31002)

This module provides an in-depth study of differential equations. You will recap techniques for solving first-order ordinary differential equations (ODEs) and explore methods for solving second-order linear ODEs. You will learn how ODEs of any order can be recast as a large first-order system and engage with techniques applicable to linear systems. In the process, you will see in practice how powerful the eigenvalues and eigenvectors of linear algebra can be. You will study the representation of general functions as a superposition of various harmonics, known as Fourier series, which arise in treating linear partial differential equations (PDEs), i.e. equations involving both space and time. 

Indicative content 

• First-order differential equations 

Separable equations, linear equations with constant coefficients, linear equations with variable coefficients, integrating factors, homogeneous equations, exact equations, and integrating factors. 

• Second-order differential equations 

Homogeneous equations with constant coefficients, fundamental solutions of linear homogeneous equations, linear independence and the Wronskian (including Abel's formula), reduction of order and reduction to the normal form, non-homogeneous equations, method of undetermined coefficients, and initial conditions. 

• Systems of first-order linear equations 

Transformation of an n-th-order equation to a system of n first-order equations, homogeneous linear systems with constant coefficients, fundamental sets of solutions and fundamental matrices, the Wronskian and Abel's formula, the exponential of a matrix, non-homogeneous linear systems, variation of parameters, homogeneous linear systems of two first-order equations with constant coefficients, stability and the phase plane. 

• Partial differential equations and Fourier series 

Fourier series of functions of one variable, Dirichlet's conditions, and how to determine Fourier coefficients (even/odd functions). Gibbs′ phenomena. Introduction to partial differential equations and the method of separation of variables, with application to initial and boundary value problems.