On a class of differential equations derived from plasma statistics

Abstract

A class of second-order differential equations stemming from an equation of motion for the two-particle spatial correlation function in one-component plasmas is studied. These equations contain an irregular singularity of varying order at the origin. The general form of solution is obtained which, together with the construction of asymptotic series, demonstrates that both solutions to all equations in the class are singular at the origin. Behavior removed from the origin is shown to be oscillatory or exponential depending on specifics of the equations.