Solution of the equivalence problem for the Painlevé IV equation

We solve the equivalence problem for the Painlevé IV equation, formulating the necessary and sufficient conditions in terms of the invariants of point transformations for an arbitrary second-order differential equation to be equivalent to the Painlevé IV equation. We separately consider three pairwise nonequivalent cases: both equation parameters are zero, $a=b=0$; only one parameter is zero, $b=0$; and the parameter $b\ne0$. In all cases, we give an explicit point substitution transforming an equation satisfying the described test into the Painlevé IV equation and also give expressions for the equation parameters in terms of invariants.