Application of the Jacobi method and integrating factors to a class of Painlevé-Gambier equations

In this work, we consider the motion of chain ball drawing with constant force in the frictionless surface which is a class of the Painleve–Gambier equations. We apply Jacobi's method which enables us to obtain Lagrangians of any second-order differential equation. It is comprised that the Lagrangian obtained by Musielak's method is the particular case of the many Lagrangians that can be obtained by Jacobi's method. In addition, we obtain integrating factors and first integrals for the equation in question by Ibragimov's variational derivative approach.