I would like to find the inverse of the following function
$$y=W(e^{ax+b})-W(e^{cx+d})+zx$$
where $W$ is the Lambert $W$ function and $a,b,c,d,z$ are some constants.
I asked the same question here:
http://math.stackexchange.com/questions/201634/inverse-function-of-y-weaxb-wecxdzx
I also checked some papers but couldnt decide how to proceed, though I have the following ideas:
$1$- Is it possible to write $$W(e^{f(x)})=W(e^{ax+b})-W(e^{cx+d})$$
$2$- Having $$y_1=W(e^{ax+b})+z_1x$$ and $$y_2=-W(e^{cx+d})+z_2x$$
where $z=z_1+z_2$, $y=y_1+y_2$ and $f^{-1}(y_1)$ and $f^{-1}(y_2)$ are known functions as already found. Can we say that $f^{-1}(y)=f^{-1}(y_1)+f^{-1}(y_2)$? or can we modify this idea to get something useful?
Any help will be appreciated. Thanks in advance.
