Solution of the equivalence problem for the Painlevé IV equation (Q393966)

The author focuses on the equivalence problem for the fourth Painlevé equation \(P_{IV}\). In particular, he provides necessary and sufficient conditions for an equation of the type \[ \, y''=P(x, y) + 3 Q(x, y)\,y'{} + 3 R(x, y)\,y'{}^2 + S(x, y)\,y'{}^3 \, \] to be equivalent by means of the point change of the variables \[ \, \tilde{x}=\tilde{x}(x, y)\,,\qquad \tilde{y}=\tilde{y}(x, y) \, \] to the \(P_{IV}\) equation \[ \, y''=\frac{1}{2 y}\,(y')^2 + \frac{3}{2}\,y^3 + 4 x\,y^2 + 2(x^2-a)\,y + \frac{b}{y}\,. \, \] The author gives these conditions and the corresponding transformations (in all of the three different cases: 1) \(a=b=0\); 2) \(b=0,\,a\neq 0\); 3) \(b\neq 0\) in an explicit form in the terms of the Cartan invariants.