Transforming Radical Differential Equations to Algebraic Differential Equations

TL;DR

This paper introduces an algorithmic method to convert differential equations with radical dependencies into polynomial form through rational variable changes, enabling easier analysis and solution.

Contribution

It generalizes previous reparametrization techniques to handle systems with radical dependencies in both ordinary and partial differential equations.

Findings

Successfully transforms radical differential equations into polynomial form

Preserves solutions through one-to-one correspondence

Extends existing reparametrization methods to broader classes of equations

Abstract

In this paper we present an algorithmic procedure that transforms, if possible, a given system of ordinary or partial differential equations with radical dependencies in the unknown function and its derivatives into a system with polynomial relations among them by means of a rational change of variables. The solutions of the given equation and its transformation correspond one-to-one. This work can be seen as a generalization of previous work on reparametrization of ODEs and PDEs with radical coefficients.