Invariants of a family of scalar second-order ordinary differential equations

The family of second-order equations with cubic nonlinearity in the first-order derivative is closed with respect to an arbitrary point change of variables. Algebraic and differential invariants for these equations which depend on the first-order derivatives are considered. We solve completely the equivalence problem for this family of equations in the generic case and also for degenerate types of these equations. Fifty equations having the Painlevé property, which have been classified by Painlevé and Gambier, belong to this remarkable family. Algebraic invariants of all these equations are calculated and constant invariants are listed in the paper. Invariant characterization of the fourth Painlevé equation is given. (paper)