Sakka, Ayman; Muğan, Uğurhan First-order second-degree equations related with Painlevé equations. (English) Zbl 1214.34088 Int. J. Nonlinear Sci. 8, No. 3, 259-273 (2009). Let \(F=\mathbb{C}(z)\). The authors consider the Painlevé equation \[ {{({v}')}^{2}}={{q}_{1}}{v}'+{{q}_{2}}, \]where \({{q}_{1}},{{q}_{2}}\in F[v]\), \(\deg {{q}_{1}}=2\) and \(\deg {{q}_{2}}=4\). The coefficients of the polynomials \({{q}_{1}},{{q}_{2}}\) are selected so that these equations have a general one-parameter family of solutions. This idea allows the authors to present some new families of solutions of Painlevé equations. For example, for equation PVI, such a family is characterized by the equation \[ {{z}^{2}}{{[(z-1){v}'-(v-1)]}^{2}}={{v}^{2}}\{2\alpha {{v}^{2}}-[(2\alpha +\lambda )z+2\alpha -\lambda ]v+2\gamma {{z}^{2}}+(2\alpha +\lambda -4\gamma )z+2\gamma -\lambda \}, \] where \(\lambda =2\gamma +2\delta -1\) and \(\beta =-\tfrac{1}{2}\). Reviewer: Mykola Grygorenko (Kyïv) MSC: 34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies 34A05 Explicit solutions, first integrals of ordinary differential equations Keywords:Painlevé equations; Fuchs theorem PDFBibTeX XMLCite \textit{A. Sakka} and \textit{U. Muğan}, Int. J. Nonlinear Sci. 8, No. 3, 259--273 (2009; Zbl 1214.34088)