Supplementary Material: Minimal Polynomial First Integrals for Second-Order ODEs

This record contains the supplementary material associated with the study: "Minimal Polynomial First Integrals for Second-Order ODEs". The dataset provides a complete list of test second-order ordinary differential equations together with their minimal polynomial first integrals (up to degree 3). The examples are classified into cubic, quadratic, linear, and non-integrable cases. The repository includes: • Supplementary.pdf – Contains the full list of test equations, classification, and verification details. – Includes a worked example illustrating the interpretation of MATLAB output. • minimalFI_2ndODE.m – MATLAB implementation of the proposed algorithm. – Detects minimal polynomial first integrals up to degree three. – Performs symbolic verification using the total derivative condition. – Includes Darboux factorization analysis. • README.txt – Instructions for usage and reproduction of results. The algorithm is based on a two-stage approach:(1) Quadratic search (f = 0)(2) Cubic search (f ≠ 0) This dataset serves both as a reproducibility package and as a benchmark for symbolic methods aimed at detecting polynomial first integrals of nonlinear differential equations.