ALL BINOMIAL-TYPE PAINLEVE EQUATIONS OF THE 2ND-ORDER AND DEGREE 3 ORHIGHER

In this paper we construct all rational Painleve-type differential equ ations which take the binomial form, (d2y / dx2)n = F(x, y, dy / dx), where n greater-than-or-equal-to 3, the case n = 2 having previously b een treated in Cosgrove and Scoufis [1]. While F is assumed to be rati onal in the complex variables y and y' and locally analytic in x, it i s shown that the Painleve property together with the absence of interm ediate powers of y'' forces F to be a polynomial in y and y'. In addit ion to the six classes of second-degree equations found in the aforeme ntioned paper, we find nine classes of higher-degree binomial Painleve equations, denoted BP-VII,...,BP-XV, of which the first seven are new . Two of these equations are of the third degree, two of the fourth de gree, three of the sixth degree, and two of arbitrary degree n. All eq uations are solved in terms of the first, second or fourth Painleve tr anscendents, elliptic functions, or quadratures. In the appendices, we discuss certain closely related classes of second-order nth-degree eq uations (not necessarily of Painleve type) which can also be solved in terms of Painleve transcendents or elliptic functions.