Analysis and object oriented implementation of the Kovacic algorithm

This paper gives a detailed overview and a number of worked out examples illustrating the Kovacic \cite{Kovacic86} algorithm for solving second order linear differential equation ${A(x) y"+ B(x) y' + C(x) y=0}$ where $A,B,C$ are rational functions with complex coefficients in the independent variable $x$. All three cases of the algorithm were implemented in a software package based on an object oriented design and complete source code listing given in the appendix with usage examples. Implementation used the Maple computer algebra language. The complete Kovacic package in one mpl file accompany the arXiv version of this paper. This package was then used to analyze the distribution of Kovacic algorithm cases on $3000$ differential equations