On Abel's Identity
Mehrzad Ajoodanian
TL;DR
The paper addresses Abel's identity by introducing a basis-invariant duality between the Wronskian-based Maurer–Cartan data and the coefficients of the underlying differential equation. It constructs the right Maurer–Cartan form $R = W'W^{-1}$ and shows that the coefficients $q_i$ of the characteristic polynomial of $R$ relate to the ODE coefficients $p_j$ in reverse order, i.e., $q_i = p_j$ for $i+j=n+1$. Abel's identity then follows as $p_1 = \frac{w'}{w}$ since $q_n = \det(R) = \frac{(\det W)'}{\det W}$. This dual perspective provides a gauge-theoretic flavor to classical ODE theory and clarifies how Wronskian data encodes the full differential equation through a natural decomposition.
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.