Handbook

Many real-world phenomena are described by ordinary differential equations (ODEs) for which there are no exact closed-form solutions. While numerical solutions can be found, these give limited insight into how the problem parameters influence the solution. This topic provides an advanced perspective on methods for analysing ODEs and finding approximate closed-form solutions to these. Common first- and second-order ODEs are first revised, which form the building blocks for developing approximate solutions, and dimensional analysis is introduced. The topic then considers phase-plane analysis and examines the idea of a bifurcation. Perturbation methods are introduced, including regular and singular perturbation, and methods to handle boundary layers. There is also a study of eigenvalue methods, with a discussion of Sturm–Liouville problems. Throughout, there is an emphasis on understanding the error of approximations and whether these are converging to the true solution.