Smoothness and analyticity of $f^{\mathrm{\prime }}=\exp (f^{-1})$

This paper considers some analytical and numerical aspects of the problem defined by an equation of the type $f^{\mathrm{\prime }}=\exp (f^{-1})$ with $f^{-1}$ is a composional inverse of $f$,some new analytical and numerical results are presented using RK4 and Explicit Runge kuta methods.