Integration of Second Order Linear Differential Equations in Elementary Functions, Example from Quantum-Mechanics

In the paper, special procedure for integrating a second order linear differential equation with six singular points, example of a quantum-mechanical problem for a spin zero particle with intrinsic Darwin-Cox structure, is considered. The method is based on the use of the Schwarzian derivative, conditions for parameters are derived at which solutions of the equation under consideration are constructed in terms of elementary functions. The graphs for several solutions are presented, their physical sense is discussed.