System of equations $x^2=y^3, x^y=y^x$ Solve the system of equations $x^2=y^3, x^y=y^x$ in positive real numbers.
Taking $\ln$ of the second equation, we have $\ln x/x=\ln y/y$. This function is increasing in $(0,e)$ and decreasing in $(e,\infty)$. For any value of $x\neq e$, we can find a unique value of $y$ such that $x^y=y^x$. But how can we find a closed form to substitute into $x^2=y^3$?

Write $y=x^{\frac{2}{3}}$, then
$$x^{x^{\frac 23}}=x^{\frac{2x}{3}}$$
so
$$x^{\frac{2}{3}}=\frac{2x}{3}$$
Can you continue?
