When mathematical modelling is used to describe physical, biological, chemical or other phenomena, one of the most common results is either a differential equation or a system of differential equations, which, together with appropriate boundary and/or initial conditions, describe the situation. These differential equations can be either ordinary (ODEs) or partial (PDEs) and finding and interpreting their solution lies at the heart of applied mathematics. This module develops the theory of differential equations with a particular focus on techniques of solving both linear and nonlinear ODEs. Fourier series, which arise in the representation of periodic functions, and special functions, which arise in the solution of PDEs such as Laplace’s equation that models the flow of potential, are also introduced. A number of the classical equations of mathematical physics are solved.