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The local equivalence problem for $d^2 y/dx^2=F(x,y,dy/dx)$ and the Painlevé transcendents
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 = Φ(x), y = ψ(x, y).There are three basic invariants which vanish iff the equation is equivalent to d 2 y/dx 2 = 0. We show that …