On the Linearization of the Third Painleve Equation

Abstract

Painleve and his school [1], [2], classified all the equations of the form y tt = F(y t , y, t) where F is rational in y t , algebraic in y and analytic in t, which have the Painleve property, i.e. their solutions are free from the movable critical points. They showed that, upto Mobius transformations, there are fifty such equations [3]. The most interesting of the fifty equations are those who are irreducible (that is, not replacable by a simpler equation or combination of simpler equations), and serve to define new transcendents. These irreducible six equations are called Painleve equations, PI-PVI.