Deciding the existence of rational general solutions for first-order algebraic ODEs. (July 2018)

• Record Type: Journal Article
• Authors: Vo, N. Thieu; Grasegger, Georg; Winkler, Franz
• Is Part Of: Journal of symbolic computation. Volume 87(2018)
• Page Start: 127
• Page End: 139
• Publication Date: 2018-07
• Subjects: 34A05 -- 34A26 -- 34A34 -- 14H45 -- 14J26 -- 68W30; Ordinary differential equation -- Algebraic curve -- Rational parametrization -- Rational general solution; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2017.06.003
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: In this paper, we consider the class of first-order algebraic ordinary differential equations (AODEs), and study their rational general solutions. A rational general solution contains an arbitrary constant. We give a decision algorithm for finding a rational general solution, in which the arbitrary constant appears rationally, of the whole class of first-order AODEs. As a byproduct, this leads to an algorithm for determining a rational general solution of a class of first-order AODE which covers almost all first-order AODEs from Kamke's collection. The method is based intrinsically on the consideration of the AODE from a geometric point of view. In particular, parametrizations of algebraic curves play an important role for a transformation of a parametrizable first-order AODE to a quasi-linear differential equation.