Solve For It

[Ben-CS]

Problem 1:

e^-x = x

Solve for x.


Problem 2:

x^y - y^x = xy - x - y

Solve for y.

[HallsofIvy]

Neither of these have "elementary" solutions (solutions in terms of the "elementary" functions: polynomial, rational, trig, log, exponential).

The first can be solved numerically or they can be solved in terms of the "Lambert W function".

[Graeme]

Problem 1:

e^-x = x
e^x = 1/x
x e^x = 1
x = ProductLog(1), where ProductLog (a.k.a. Lambert's W-function) is defined as the inverse of f(W) = W e^W

Numerically, the answer is about 0.567143290409784

Problem 2:

I don't know. Sorry.