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.