On the Asymptotic Behavior of the Solutions of Nearly Regular Singular Differential Equations

In this paper we consider families of differential equations on some real interval that depend on a parameter ε ε , are regular for ε ≠ 0 ε ≠ 0 and have exactly one regular singular point for ε =0 ε = 0 . Our aim is to give a formula which describes the asymptotic behavior of the solutions for ε → 0 ε → 0 . To this end, we use a method that combines matched asymptotic analysis with uniform asymptotic integration. We then apply our results to some typical examples, such as a differential equation with two coalescing regular singular points as well as a singular perturbation of the Bessel equation.