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On a transformation of the Painlevé III and V equations. (Russian. English summary) Zbl 0845.34009
Summary: We apply an analytic method to III and V Painlevé equations. This method reduces the Painlevé equations to an equation of the form ${d^2 w\over dv^2}= K(w, v)^{- 1}\Biggl( E(w, v)\Biggl({dw\over dv}\Biggr)^3+ F(w, v)\Biggl({dw\over dv}\Biggr)^2+ G(w, v) {dw\over dv}+ H(w, v)\Biggr),$ where $$E$$, $$F$$, $$G$$, $$H$$, $$K$$ are polynomials of $$w$$, $$v$$; $$w= w(z)$$, $$v= v(z)$$ are analytic functions of $$z$$ at the field $$\mathfrak D$$.
##### MSC:
 34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
##### Keywords:
analytic method; III and V Painlevé equations