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First-order second-degree equations related with Painlevé equations. (English) Zbl 1214.34088

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 \(\beta =-\tfrac{1}{2}\).

MSC:

34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
34A05 Explicit solutions, first integrals of ordinary differential equations
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