A multivariable real analysis problem from Pugh. Here is an exercise from Pugh real analysis which i can't solve:
Consider the equation $xe^y + ye^x = 0$
Prove that there is no way to write down an explicit solution $y=y(x)$ in a nbhd of origin.
I think implicit function theorem could be useful but  I've made a mistake: if we calculate the derivation relative to 'y'  we obtain: $x e^y + e^x $ & near origin it equals 1 !
so by implicit function theorem an explicit solution must be found!!
Where am I wrong?
Thanks
 A: You are right: The function
$$f(x,y):=xe^y+ye^x$$
has gradient  $\nabla f(0,0)=(1,1)$. Therefore the equation $f(x,y)=0$ implicitly defines functions $y=\phi(x)$ and $x=\psi(y)$,  defined in a neighborhood of $x=0$, resp., $y=0$. For symmetry reasons one has in fact $\phi=\psi$, and as $\psi$ is the inverse of $\phi$ this implies that $\phi$ is an involution.
The function $\phi$ is  even analytic in a neighborhood of $0$ and has a convergent Taylor expansion there. Computation gives
$$\phi(x)=-x + 2 x^2 - 4 x^3 + {28\over3} x^4 - 24 x^5 + {328\over5} x^6 - {8416 \over45}x^7+\ldots$$
I think the statement in your source is meant in the sense that this function $\phi$ cannot be expressed in terms of elementary functions. To really prove such a statement is terribly difficult.
A: The precise words of Charles Pugh are: 

Observe that there is no way to write down an explicit solution $y=y(x)$ of (25) in a neighborhood of the point $(x_0,y_0)=(0,0)$.

You are invited to remark that it is impossible to solve the equation with respect to $y$ in terms of elementary functions. I agree that this is either a stupid question (everybody should answer that it is impossible) or a formidable question (nobody could rigorously prove that there is no way to find $y$ in terms of elementary functions, at least no reader of this book).
