Extended airy function and differential equations with n-turning points

Abstract

This paper studies a second order linear ordinary differential equation with n-turning points

$$\frac{{d^2 y}}{{dx^2 }} + [\lambda ^2 q_1 (x) + q_2 (x)]y = 0$$

Where q₁(x)=(x-μ₁) (x-μ₂) ... (x-μ_m) f(x), f(x)≠0, and λ is a large parameter.

The formal uniformly valid asymptotic solution of the equation is obtained based on the analysis of the three points by means of the matched method. By the work a method is developed and the applicability of this method to the n-turning points is demonstrated.