Ordinary differential equations, and second-order equations in particular, are at the heart of many mathematical descriptions of physical systems, as used by engineers, physicists and applied mathematicians. This text provides an introduction to all the relevant material normally encountered at university level: series solution, special functions (Bessel, etc.), Sturm-Liouville theory (involving the appearance of eigenvalues and eigenfunctions) and the definition, properties and use of various integral transforms (Fourier, Laplace, etc.). Numerous worked examples are provided throughout.