Existence and convergence of Puiseux series solutions for autonomous first order differential equations. (January 2022)

• Record Type: Journal Article
• Authors: Cano, José; Falkensteiner, Sebastian; Sendra, J. Rafael
• Is Part Of: Journal of symbolic computation. Volume 108(2022)
• Page Start: 137
• Page End: 151
• Publication Date: 2022-01
• Subjects: Algebraic differential equation -- Algebraic curve -- Place -- Formal Puiseux series solution -- Convergent 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.2020.06.010
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: Given an autonomous first order algebraic ordinary differential equation F ( y, y ′ ) = 0, we prove that every formal Puiseux series solution of F ( y, y ′ ) = 0, expanded around any finite point or at infinity, is convergent. The proof is constructive and we provide an algorithm to describe all such Puiseux series solutions. Moreover, we show that for any point in the complex plane there exists a solution of the differential equation which defines an analytic curve passing through this point.