An inverse function and a differential equation What is the inverse function of $f(x)=x+\exp(x)$?
I doubt it's the solution of the differential equation: $$y'+y'\exp(y)-1=0.$$

$$y'(1+e^y)=1$$
Integration, with constant $c$ :
$$y+e^y=x+c$$
$$x+c-y=e^y$$
$$(x+c-y)e^{-y}=1$$
$$(x+c-y)e^{(x+c-y)}=e^{x+c}$$
Let $Y=(x+c-y)$ and $X=e^{x+c}$
$$Ye^Y=X$$
With the Lambert W function :
$$Y=W(X)$$
$$x+c-y=W\left(e^{x+c}\right)$$
$$y=x+c-W\left(e^{x+c}\right)$$
