  we show that the solution to an oscillatory-singularly perturbed ordinary
differential equation may be asymptotically expanded into a sum of oscillating
terms. Each of those terms writes as an oscillating operator acting on the
solution to a non oscillating ordinary differential equation with an
oscillating correction added to it. The expression of the non oscillating
ordinary differential equations are defined by a recurrence relation. We then
apply this result to problems where charged particles are submitted to large
magnetic field.
