Generalized multitime expansions for equations with slowly varying coefficients

Abstract

The successive terms in a uniformly valid multitime expansion of the solutions of constant coefficient differential equations containing a small parameter ϵ may be obtained without resorting to secularity conditions if the time scales ti=ϵit(i=0,1,…) are used. Similar results have been achieved in some cases for equations with variable coefficients by using nonlinear time scales generated from the equations themselves. This paper extends the latter approach to the general second order ordinary differential equation with slowly varying coefficients and examines the restrictions imposed by the method.

Keywords

• ordinary differential equations
• asymptotic expansions
• perturbations.