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• W2198179308 abstract "In this paper we construct all rational Painlevé-type differential equations which take the binomial form, (d2y/dx2)n = F(x,y,dy/dx), where n ≥ 3, the case n = 2 having previously been treated in Cosgrove and Scoufis [1]. While F is assumed to be rational in the complex variables y and y′ and locally analytic in x, it is shown that the Painlevé property together with the absence of intermediate powers of y″ forces F to be a polynomial in y and y′. In addition to the six classes of second-degree equations found in the aforementioned paper, we find nine classes of higher-degree binomial Painlevé 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 degree, three of the sixth degree, and two of arbitrary degree n. All equations are solved in terms of the first, second or fourth Painlevé transcendents, elliptic functions, or quadratures. In the appendices, we discuss certain closely related classes of second-order nth equations (not necessarily of Painlevé type) which can also be solved in terms of Painlevé transcendents or elliptic functions." @default.
• W2198179308 created "2016-06-24" @default.
• W2198179308 creator A5074183514 @default.
• W2198179308 date "1993-11-01" @default.
• W2198179308 modified "2023-10-10" @default.
• W2198179308 title "All Binomial-Type Painlevé Equations of the Second Order and Degree Three or Higher" @default.
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• W2198179308 doi "https://doi.org/10.1002/sapm1993902119" @default.
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