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

This paper addresses necessary and sufficient factorizability condi- tions for classes of second order linear ordinary differential equations (ODEs) characterized by the degrees of their corresponding polynomial functions coef- ficients. 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 differ- ential operator. Concrete examples are probed, including special cases of Bocher ODEs like Heun, extensions of Wangerin and Heine's differential equations. AMS 2010 Mathematics Subject Classification. 26C15, 34-04, 34A05, 34A30, 47E05.