Is there analytic solution to $x^y=y^x\land x\neq y$ as $y(x)$?

Question

Equation $x^y=y^x\land x\neq y$ has trivial solution $ y(x) = x$. Is there non trivial solution given say in terms of elementary or special functions as $y(x)$? A solution that would yield $y(2) = 4$ and $y(4) = 2$ - among of course all other real numbers $x > 0$.

Answer 1

I already answered this question here in the 3rd comment. If $W(t)$ is Lambert W function, then solution is

$ y(x) = \begin{cases} e^{-W_{-1}\left(-\log \left(x^{\frac{1}{x}}\right)\right)} & 0<x<e \\ e^{-W\left(-\log \left(x^{\frac{1}{x}}\right)\right)} & e<x \end{cases} $

With the graph that looks like (so no trivial $x = y$ part present):

Answer 2

if you take the $xy$-th root of your equation, it yields:

$$x^{\frac1x} = y^{\frac1y} $$

Answer 3

As far as I remember Lambert's W function can solve some of such equations. If I'm not mistaken, Euler was the first person who studied the solutions of this equation in details and he wrote a treatise on Lambert's W function with his work toward an infinite series that converged to the Lambert W function in a restricted radius. His work possibly motivated Lagrange's inversion formula. I guess nowadays Lambert's W function has been studied well and you might find much more information regarding it.

Answer 4

There are no non-trivial and non-elementary solutions to the functional equation $x^{y(x)} \equiv y(x)^x$.

The so-called Lambert W-function can be used to describe the solution.

Besides the trivial $y=x$ solution, there is also the following:

$$y(x) = -\frac{x\operatorname{LambertW}\left(-\frac{\ln x}{x}\right)}{\ln x}$$

I asked Maple to plot the solutions to $x^y=y^x$ in the range $0 \le x, y \le 10$ and got the plot attached below. The diagonal line is the trivial solution $y=x$. The other is the LambertW solution.

Answer 5

The function $f(x)=x^{1/x}$ is strictly increasing, in $(0,\mathrm{e})$, as $$ f'(x)=x^{1/x}\left(\frac{1-\ln x}{x^2}\right)>0, $$ and strictly decreasing in $(\mathrm{e},\infty)$, with $$ \lim_{x\to 0}x^{1/x}=0 \quad \text{and} \quad \lim_{x\to\infty}x^{1/x}=1. $$ In particular, every value in the interval $(1,\mathrm{e}^{1/\mathrm{e}})$ is taken twice by $f$, i.e., for all $a\in(1,\mathrm{e}^{1/\mathrm{e}})$, there exist $x,y$, such that $$ 1<x<\mathrm{e}<y \quad\text{and}\quad x^{1/x}=y^{1/y}, $$ and thus $x^y=y^x$. Let's see now how can we express $y=y(x)$. Let $f(x)=x^{1/x}$, where $x\in(1,\mathrm{e})$. Then, the unique $y\in(\mathrm{e},\infty)$, with the property that $f(x)=f(y)$ is equal to: $$ y=\left(\left(\,f\restriction_{(\mathrm{e},\infty)} \right)^{-1}\circ f\right)(x). $$
The function: $$ g=\left(\,f\restriction_{(\mathrm{e},\infty)} \right)^{-1}\circ f :(1,\mathrm{e})\to (\mathrm{e},\infty), $$ is clearly 1-1 and onto, strictly decreasing, and $C^\infty$, and satisfies $$ x^{g(x)}=\big(g(x)\big)^x \quad\text{and}\quad x<\mathrm{e}<g(x). $$