Uniform asymptotic solution of second order linear differential equations without turning varieties

Abstract:

The purpose of this paper is to initiate the study of a new kind of asymptotic series expansion for solutions of differential equations containing a parameter. We obtain uniform asymptotic solutions for certain equations of the form \[ \epsilon^{2n} y'' = a(t, \epsilon )y, \quad (\quad)' = d/dt, \] where $n$ is a positive integer, $t$ and $\epsilon$ are real variables ranging over $\left | t \right | \leqq {t_0},0 < \epsilon \leqq { \epsilon _0}$, and $a$ is a function infinitely differentiable on the closure of this domain. We require that $a(t, \epsilon )$ satisfy conditions which can be regarded as generalized nonturning-point conditions. These conditions imply the absence of secondary turning points, and reduce in the simplest case to the condition $a(t,0) \ne 0$, but also include cases (the interesting ones) in which $a(0,0) = 0$