Existence and convergence of Puiseux series solutions for autonomous first order differential equations

TL;DR

This paper proves that all formal Puiseux series solutions of autonomous first order algebraic differential equations are convergent and provides an algorithm to explicitly find all such solutions, ensuring solutions pass through any given point.

Contribution

It establishes convergence of Puiseux series solutions and introduces a constructive algorithm to describe all solutions for autonomous first order algebraic ODEs.

Findings

All formal Puiseux series solutions are convergent.

An explicit algorithm to find all solutions is provided.

Solutions can pass through any point in the complex plane.

Abstract

Given an autonomous first order algebraic ordinary differential equation F(y,y')=0, we prove that every formal Puiseux series solution, 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.