From formal to actual Puiseux series solutions of algebraic differential equations of first order

TL;DR

This paper investigates the existence, uniqueness, and convergence of formal Puiseux series solutions for first-order algebraic differential equations, especially when classical Painleve theorem conditions are not met, supported by illustrative examples.

Contribution

It extends the understanding of Puiseux series solutions by analyzing cases where Painleve theorem does not apply, including new examples and connections to Petrovic's results.

Findings

Established conditions for existence and uniqueness of solutions.

Analyzed convergence in cases beyond Painleve theorem applicability.

Provided illustrative examples linking to Painleve and Petrovic's results.

Abstract

The existence, uniqueness and convergence of formal Puiseux series solutions of non-autonomous algebraic differential equations of first order at a nonsingular point of the equation is studied, including the case where the celebrated Painleve theorem cannot be applied explicitly for the study of convergence. Several examples illustrating relationships to the Painleve theorem and lesser-known Petrovic's results are provided.