<s>[INST] 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 $x=y=0$
 [/INST] Note that if $x^2=y^3$, then $y=x^{2/3}$.  Therefore, we find that 
$$x^{x^{2/3}}=\left(x^{2/3}\right)^x \tag 1$$
Taking the logarithm of both sides of $(1)$ yields
$$x^{2/3}\log(x)=\frac23 x\log(x) \tag2$$
Solutions to $(2)$ are $x=1$ and $x=27/8$.  For $x=1$, $y=1$ and for $x=27/8$, $y=9/4$.
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