Equivalence of ordinary differential equations y″ = R(x, y)y′ 2 + 2Q(x, y)y′ + P(x, y)

Open Problems for Painlevé Equations

“…where F is rational in w and dw/dz and locally analytic in z and solved the equations in terms of the first, second and fourth Painlevé transcendents, elliptic functions, or quadratures. For various results on classifying classes of second-order ordinary differential equations, including Painlevé equations, see Babich and Bordag [12], Bagderina [13,15,16,17,18], Bagderina and Tarkhanov [19], Berth and Czichowski [22], Hietarinta and Dryuma [78], Kamran, Lamb and Shadwick [88], Kartak [90,91,92,93], Kossovskiy and Zaitsev [97], Milson and Valiquette [106], Valiquette [139] and Yumaguzhin [145]. Most of these studies are concerned with the invariance of second-order ordinary differential equations of the form…”

“Painlevé 34” equation: Equivalence test

“…To simplify the notation below we do not write the tildes over the variables x and y in the equation (20). This equation also has form (5) with the coefficients P = −β 2 y 1/3 6y + 3xy 2/3 + 3 2 , Q = 0, R = 5 18y…”

“…transforms equation (2) (that is written in variables (x, y)) into equation (20) (that is written into variables (x,ỹ)) with the parameter β 2 = 4a 2 .…”

“…is equivalent to "Painlevé 34" equation (20) with the parameter β 2 = 2k 2 ν 2 1 k 2 1 /C 2 if ν 1 = 0, k 1 = 0, k 2 = 0, C = 0. All conditions of Theorem 2 are true.…”

“…transforms this equation (that is written in variables (x, w)) into the equation (20) (that is written into variables (x,ỹ)). And it is equivalent to Painlevé II equation (1) with the parameter a = 0 if ν 1 = 0, k 1 = 0, k 2 = 0, C = 0.…”