On Abel's Identity

TL;DR

This paper introduces a duality linking the coefficients of the characteristic polynomial of the Maurer-Cartan form of the Wronskian matrix to those of the original differential equation, deriving Abel's identity as a consequence.

Contribution

It establishes a novel duality framework that connects differential equation coefficients with Maurer-Cartan forms, providing a new perspective on Abel's identity.

Findings

Derived a duality matching polynomial coefficients

Revealed a new interpretation of Abel's identity

Connected Maurer-Cartan forms with differential equations

Abstract

We provide a natural duality that matches, in reverse order, the coefficients of the characteristic polynomial of the Maurer-Cartan of the Wronskian matrix with the coefficients of the original differential equation. Abel's identity is recovered as a corollary.