How to Solve Exponential and Polynomial Equations?

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The discussion addresses two mathematical problems that lack elementary solutions. For the first problem, e^-x = x, it can be solved using the Lambert W function, leading to the solution x = ProductLog(1), with a numerical approximation of approximately 0.567143290409784. The second problem, x^y - y^x = xy - x - y, remains unsolved, with no clear method provided for finding y. The focus is on the limitations of elementary functions in solving these equations.
Ben-CS
Problem 1:

e^-x = x

Solve for x.


Problem 2:

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

Solve for y.
 
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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".
 
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.
 

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