Solution of $x^y=y^x$ and $x^2=y^3$ Solve the given set of equations:
$x^y=y^x$ and $x^2=y^3$ where $x,y \in \mathbb{R}$
Would any other solution exist other that $x=y=1$ because I think $x^2=y^3$
will only be true for $x=y=1$ or $x=y=0$

Write $x=y^{3/2}$ and substitute in the first equation to get $y^{3y/2}=y^{y^{3/2}}$; if $y \neq 0,1$, then the exponents must be equal so that $3y/2=y^{3/2}$, which gives $y=\dfrac{9}{4}$ and $x=\dfrac{27}{8}$.
