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.本数据集为研究论文《二阶常微分方程的极小多项式首次积分》（Minimal Polynomial First Integrals for Second-Order ODEs）的配套补充材料。 本数据集完整收录了全部测试用二阶常微分方程，及其不超过三次的极小多项式首次积分。所收录的示例被划分为三次、二次、线性及不可积四类情形。 本附属仓库包含以下内容： • Supplementary.pdf —— 收录了全部测试方程、分类结果与验证细节；同时附带一个演示案例，用于阐释MATLAB输出结果的解读方法。 • minimalFI_2ndODE.m —— 本文所提算法的MATLAB实现脚本，可检测不超过三次的极小多项式首次积分；基于全微分条件执行符号验证，并包含达布（Darboux）因式分解分析模块。 • README.txt —— 包含使用说明与结果复现步骤。 本算法采用两阶段框架：(1) 二次搜索阶段（f = 0）；(2) 三次搜索阶段（f ≠ 0）。 本数据集既可作为可复现研究的配套套件，亦可作为面向非线性微分方程多项式首次积分检测的符号方法基准测试集。