How to analytically solve for $x$ in $x = e^{-1/x}$?

Question

As the title goes, how do we analytically solve for $x$ in $x = e^{-1/x}$?

I attempted to differentiate then integrate and assumed c = 0, i get x = -2, but I am doubtful.

What would be the exact approach to solve this analytically?

Answer 1

Write

$$-\frac1xe^{-1/x}=-1$$

and

$$-\frac1x=W(-1).$$

Answer 2

Substitute $x\to -1/z$ we get $-1/z=e^z$ hence $-1=ze^z$ hence $z=W(-1)$ hence $$ x = -\frac1{W(-1)}$$ Where $W$ is Lambert's function W.

The substitution is in order because $x=0$ is not a solution.