Analysis and object oriented implementation of the Kovacic algorithm

TL;DR

This paper provides a comprehensive overview and an object-oriented implementation of Kovacic's algorithm for solving second order linear differential equations with rational function coefficients, including software and analysis of case distribution.

Contribution

It introduces an object-oriented software package implementing Kovacic's algorithm in Maple, with detailed examples and case analysis on numerous differential equations.

Findings

Implemented Kovacic's algorithm in Maple with complete source code.

Analyzed the distribution of algorithm cases across 3000 differential equations.

Provided worked-out examples illustrating the algorithm's application.

Abstract

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