Uniqueness of local, analytic solutions to singular ODEs

TL;DR

This paper establishes the existence and uniqueness of local analytic solutions for a class of singular ODEs, ensuring smoothness at critical points, with applications to modeling axially symmetric surfaces.

Contribution

It introduces a novel approach translating singular initial value problems into regular ODE equilibrium problems, enabling classical invariant manifold techniques.

Findings

Proves existence and uniqueness under non-resonance conditions

Demonstrates smoothness at the tip of axially symmetric surfaces

Applies invariant manifold theory to singular ODEs

Abstract

We study local, analytic solutions for a class of initial value problems for singular ODEs. We prove existence and uniqueness of such solutions under a certain non-resonance condition. Our proof translates the singular initial value problem to an equilibrium problem of a regular ODE. Then, we apply classical invariant manifold theory. We demonstrate that the class of ODEs under consideration captures models which describe the shape of axially symmetric surfaces which are closed on one side. Our main result guarantees smoothness at the tip of the surface.