# First integrals in the Brans-Dicke cosmology

## TL;DR

This paper identifies first integrals, Darboux polynomials, and Jacobi's last multipliers in Brans-Dicke cosmology, providing algebraic invariant sets and visualizations for specific parameter values, advancing the understanding of its mathematical structure.

## Contribution

It introduces the first and second degree Darboux polynomials, Jacobi's last multipliers, and first integrals specific to Brans-Dicke cosmology, which were not previously detailed.

## Key findings

- Derived first integrals for specific $$ parameter values.
- Constructed algebraic invariant sets.
- Visualized first integrals for particular $$ values.

## Abstract

The work presents the frst and the second degree Darboux polynomials, Jacobi's last multipliers as well as the set of frst integrals for Brans-Dicke cosmology. Algebraic invariant sets are constructed. First integrals are visualized for some particular values of the $\omega$ parameter.