How to Solve Exponential and Polynomial Equations?

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SUMMARY

This discussion focuses on solving exponential and polynomial equations, specifically the equations e^-x = x and x^y - y^x = xy - x - y. The first equation can be solved using the Lambert W function, where the solution is expressed as x = ProductLog(1), yielding a numerical approximation of 0.567143290409784. The second equation does not have an elementary solution, and the author admits uncertainty regarding its resolution.

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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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