General Prerequisites: It is recommended to take Integral Transforms in parallel with Differential Equations 2.
Course Overview: This course continues the Differential equations 1 course, with the focus on boundary value problems. The course aims to develop a number of techniques for solving boundary value problems and for understanding solution behaviour. The course concludes with an introduction to asymptotic theory and how the presence of a small parameter can affect solution construction and form.
Learning Outcomes: Students will acquire a range of techniques for solving second order ODE's and boundary value problems. They will gain a familiarity with ideas that are applicable beyond the direct content of the course, such as the Fredholm alternative, Bessel functions, and asymptotic expansions.
Course Synopsis: Models leading to two point boundary value problems for second order ODEs
Inhomogeneous two point boundary value problems (\(Ly=f\)); Wronskian and variation of parameters. Green's functions.
Adjoints. Self-adjoint operators. Eigenfunction expansions (issues of convergence and completeness noted but full treatment deferred to later courses). Sturm-Liouville theory. Fredholm alternative.
Series solutions. Method of Frobenius. Special functions.
Asymptotic sequences. Approximate roots of algebraic equations. Regular perturbations in ODE's. Introduction to boundary layer theory.
