Classes of Painlevé type second-order differential equations of the second degree (Q5926574)

Here, the authors consider the following equation \[ \left(y''-\left( 1-{1\over n}\right){y'}^2/y+ A(x,y)y'+ B(x,y)\right)^2 =F(x,y)\tag{1} \] where \(A(x,y)\), \(B(x,y)\) and \(F(x,y)\) are rational functions in \(y\) with analytic coefficients in \(x,n\) is an integer other than 0 and \(\infty\). The authors prove that if equation (1) has the Painlevé property, then it reduces into one of the thirteen typical equations, and the solutions to these equations can be expressed via either the solutions to the first four canonical Painlevé equations, elliptic functions or solutions linear equations.