Prove $xe^y + ye^x =0$ has no explicit solution $y(x)$

Question

By the implicit function theorem there exists a solution $ y(x)$ of $$xe^y + ye^x=0$$ in a neighborhood of $ (0,0)$. Nevertheless, my textbook asks me to "observe that there is no way to write down an explicit solution $ y=y(x)$ in a neighborhood of the point $ (x_0, y_0) =(0,0)$." How can I show this?

Answer 1

There is no elementary function, but using the Lambert W function, we get $$ y=-\operatorname{W}(xe^{-x}) $$