Ring of flows of one-dimensional differential equations

TL;DR

This paper constructs and analyzes the ring of flows for autonomous first-order differential equations over integral domains, exploring its structure and solutions via formal exponential generating series.

Contribution

It introduces the autonomous ring of flows and studies its structure, providing a framework for solving differential equations through formal series decomposition.

Findings

Built the autonomous ring of flows for first-order differential equations

Analyzed the structure of the ring of formal exponential generating series

Provided methods to find solutions when the vector field decomposes

Abstract

In this article, the ring of flows of autonomous differential equations of order one on integral domains is constructed. First, we build the autonomous ring $\Opa(\hurw_{R}[[x]])$ and then its structure is studied. Next, we build the ring of formal exponential generating series of the ring $\Opa(\hurw_ {R} [[x]]) $, where it is possible to find solutions of differential equations of order one when the vector field of the system can be decomposed in sum or product of functions.