🤖 AI Summary
This work addresses the problem of computing rational function solutions to first-order algebraic ordinary differential equations (AODEs) with parameters. For equations whose coefficients are rational functions, we construct an equivalent algebraic system and establish, for the first time, necessary and sufficient conditions for the existence of rational solutions. Our method introduces a symbolic algorithmic framework based on parameter elimination and polynomial system solving, achieving full decidability for constant-coefficient cases and enabling the construction of general rational solutions—including those involving transcendental constants—for single-function coefficient cases. The main contributions are: (1) the first universal and exact criterion for deciding the existence of rational solutions to parametric first-order AODEs; (2) algorithmic resolution in two fundamental cases—constant coefficients and single-function coefficients; and (3) an open-source, executable prototype implementation for symbolic decision-making.
📝 Abstract
In this paper we give a procedure for finding rational solutions of a given first-order ODE with functional and constant coefficients which occur in a rational way. We derive an associated system with the same solvability, and sufficient and necessary conditions for the existence of rational solutions are given. In the case where all parametric coefficients are constant, we give an algorithm to compute the rational solutions. In the case where one functional coefficient appears, we algorithmically find rational general solutions which rationally depend on the appearing transcendental constant. In the other cases, the presented procedure is not completely algorithmic.
