PAINLEVE CLASSIFICATION OF A CLASS OF DIFFERENTIAL-EQUATIONS OF THE 2ND-ORDER AND 2ND DEGREE

In this paper we construct all Painleve-type differential equations of the form (d2y / dx2)2 = F(x, y, dy / dx), where F is rational in y and y' = dy / dx, locally analytic in x, and not a perfect square. No further simplifying assumptions are made, but it is found that the absence of a term linear in y'' in the class of equations under investigation forces F to be a polynomial in y and y'. We find exactly six distinct classes of second-degree Painleve equations, denoted SD-I,...,SD-VI, some of which further subdivide into canonical subcases. Only the first three classes (or at least equations transformable to the first three classes) and part of the sixth have appeared previously in the literature, especially the work of Chazy and Bureau. The fourth and fifth classes are new. The unified treatment of SD-I, which we call the ''master Painleve equation,'' is new. Complete solutions are given in terms of the classical Painleve transcendents, elliptic functions, or solutions of linear equations. In an appendix, it is shown that a class of second-degree equations generalizing the Appell equation can always be reduced to a second-order linear equation.