Solution to y*ln(y)=x*exp(x)

Question

Can anyone please help me with a close-form explicit solution to $y\times\ln(y)=x\times\exp(x)$ ? Please note that $y=\exp(x)$ is not the answer I am looking for. The equation actually looks like this $y=\exp(x/y*\exp(x))$

Answer 1

With $y=e^t$, the equation reads

$$te^t=xe^x.$$

An obvious solution is $t=x$. Anyway, for $t<0$, the LHS function is not invertible, and you need the two branches of Lambert's function.

$$t_0=W(xe^x)=x,\\t_1=W_{-1}(xe^x).$$