Solving second order homogeneous differential equations in terms of Heun's general function

Abstract

In this paper, we present an algorithm that checks if a second-order differential operator $L\in C\left(x\right)[\partial ]$ can be reduced to the general Heun's differential operator. The algorithm detects the parameters of the transformations in $C\left(x\right)[\partial ]$ that transfer the general Heun's differential operator to the operator L whose solutions are of the form(1)$\exp (\int rdx)\cdot \mathrm{HeunG}(a,q;\alpha ,\beta ,\gamma ,\delta ;f(x\left)\right),$ where $\{\alpha ,\beta ,\delta ,\gamma \}\in Q\setminus Z$, the functions $r,f\left(x\right)\in C\left(x\right)$, and $C\left(f\right(x\left)\right)$ is a subfield of $C\left(x\right)$ of index 2 or 3 or $f\left(x\right)=\frac{a{x}^n+b}{c{x}^n+d}$ for some n in $N\setminus \left\{1\right\}$.