From curvature to Kovacic: a geometric approach to integrability of scalar ODEs

TL;DR

This paper explores a geometric approach to the integrability of scalar first-order ODEs where the surface curvature depends only on the independent variable, linking it to a second-order linear operator and applying differential Galois theory.

Contribution

It establishes a novel connection between curvature-dependent ODEs and linear operators, providing criteria for integrability via Kovacic's algorithm and Liouvillian solutions.

Findings

Solutions satisfy a Riccati equation linearized by a second-order operator

Integrability by quadratures linked to Liouvillian solutions of the linear operator

Kovacic's algorithm offers a decision procedure when curvature is rational

Abstract

We study first-order ordinary differential equations such that the intrinsic Gauss curvature of the associated surface depends only on the independent variable: $\mathcal{K}(x,u)=\kappa(x)$, showing that this geometrically motivated class of equations admits a threefold connection to the second-order linear operator $L=d^2/dx^2+\kappa(x)$: the divergence along every solution satisfies a Riccati equation that linearizes to $L(y)=0$; every solution of the first-order equation satisfies the non-homogeneous equation $L(u)=c(x)$; and solutions of $L(y)=0$ give rise to integrating factors for the original nonlinear equation. By means of differential Galois theory, we prove that the nonlinear equation is integrable by quadratures if and only if $L$ admits a non-zero Liouvillian solution; when $\kappa$ is rational, Kovacic's algorithm provides a complete decision procedure.