On factorizable classes of second order linear ordinary differential equations with rational functions coefficients

Abstract

This paper addresses necessary and sufficient factorizability conditions for classes of second order linear ordinary differential equations (ODEs) characterized by the degrees of their corresponding polynomial functions coefficients. A pure algebraic method is used to solve a system of linear algebraic equations whose solutions satisfy a compatibility criterion and generate two first order differential operators factorizing the considered second order differential operator. Concrete examples are probed, including special cases of Böcher ODEs like Heun, extensions of Wangerin and Heine’s differential equations.