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

Abstract In this paper we consider families of differential equations on some real interval that depend on a parameter $$\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> , are regular for $$\varepsilon \ne 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ε</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and have exactly one regular singular point for $$\varepsilon =0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ε</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . Our aim is to give a formula which describes the asymptotic behavior of the solutions for $$\varepsilon \rightarrow 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ε</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . 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.