Point equivalence of second-order ordinary differential equations to the fifth Painlevé equation with one and two nonzero parameters

Abstract

We consider the problem of the equivalence of scalar second-order ordinary differential equations under invertible point transformations. To solve this problem in the case of Painleve equations, we use previously constructed invariants of a family of equations whose right-hand sides have a cubic nonlinearity in the first derivative. We obtain the conditions for point equivalence to the fifth Painleve equation if two or three of its parameters are equal to zero.