The local equivalence problem for $d^2 y/dx^2=F(x,y,dy/dx)$ and the Painlevé transcendents

Abstract

We solve the local equivalence problem for d 2 y/dx 2 = F(x, y, dy/dx) under the natural group of coordinate changes x = (x), y = (x, y). There are three basic invariants which vanish iff the equation is equivalent to d 2 y/dx 2 = 0. We show that two of the invariants vanish for the six Painleve transcendents and that the third can be used to produce a complete set of invariants. We give necessary and sufficient conditions for d 2 y/dx 2 = F(x,y, dy/dx) to be equivalent to either of the first two Painleve transcendents and give simple algebraic formulas for the change of variable which puts an equivalent equation into the standard form.