# A6: Differential Equations 2 (2016-2017)

It is recommended to take Integral Transforms in parallel with Differential Equations 2.

16 lectures

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.

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.

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.

K. F. Riley, M. P. Hobson and S. J. Bence, *Mathematical Methods for Physics and Engineering*, (3rd Ed. Cambridge University Press, 2006).

W. E. Boyce & R. C. DiPrima, *Elementary Differential Equations and Boundary Value Problems* (7th edition, Wiley, 2000).

P. J. Collins, *Differential and Integral Equations* (O.U.P., 2006).

Erwin Kreyszig, *Advanced Engineering Mathematics* (8th Edition, Wiley, 1999).

E. J. Hinch, *Perturbation Methods* (Cambridge University Press, Cambridge, 1991).

J. D. Logan, *Applied Mathematics*, (3rd Ed. Wiley Interscience, 2006).