  We present a methodology for numerically integrating ordinary differential
equations containing rapidly oscillatory terms. This challenge is distinct from
that for differential equations which have rapidly oscillatory solutions: here
the differential equation itself has the oscillatory terms. Our method
generalises Filon quadrature for integrals, and is analogous to integral
techniques designed to solve stochastic differential equations and, as such, is
applicable to a wide variety of ordinary differential equations with rapidly
oscillating factors. The proposed method flexibly achieves varying levels of
accuracy depending upon the truncation of the expansion of certain integrals.
Users will choose the level of truncation to suit the parameter regime of
interest in their numerical integration.
