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arXiv · 2604.05752

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

A. J. Pan-Collantes·J. A. Álvarez-García
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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.

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A. J. Pan-Collantes, J. A. Álvarez-García. 2026-04-07. From curvature to Kovacic: a geometric approach to integrability of scalar ODEs. https://arxiv.org/abs/2604.05752

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