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arXiv · 2610.32769

Rational $S$-functions for second order ODEs with non-Liouvillian first integrals

L. G. S. Duarte·L. A. C. P. da Mota
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Abstract

The $S$-function method for rational second order ODEs $y''=φ(x,y,y')$ uses $S=I_y/I_z$, attached to a first integral $I$, which satisfies a Riccati equation along the flow; it was designed for Liouvillian $I$. We study the case of rational $S$ with non-Liouvillian $I$. From any rational $g(x,y,z)$ with $g_z\neq0$ and any linear equation $A_2w''+A_1w'+A_0w=0$ with polynomial coefficients, the substitution $w'/w=g(x,y,y')$ yields a rational 2ODE whose first integral is built from two independent solutions, and $S=g_y/g_z$: the Wronskian cancels, so $S$ does not see the linear equation, and one rational $S$ is shared by equations with and without Liouvillian first integrals. Conversely, given $(φ,S)$, membership in this class reduces to a rational first integral of an explicit planar vector field plus a Möbius-invariant Riccati test in $x$, which recovers the linear equation. A transfer theorem shows that $I$ is Liouvillian if and only if this equation has a Liouvillian solution, so Liouvillianity is decidable on the class by Kovacic's algorithm: a three-variable problem with no a priori degree bound is reduced to two classical ones. On nine pull-back equations of degree up to $15$, given without information on their construction, the procedure recovers the Airy equation and the Bessel equation of order zero.

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BibTeXRIS

L. G. S. Duarte, L. A. C. P. da Mota. 2026-10-23. Rational $S$-functions for second order ODEs with non-Liouvillian first integrals. https://arxiv.org/abs/2610.32769

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