Solve xy^m=yx^3

$$x y ^ { m } = y x ^ { 3 }$$

Solve for m (complex solution)

$$\left\{\begin{matrix}m=\log_{y}\left(yx^{2}\right)+\frac{2\pi n_{1}i}{\ln(y)}\text{, }n_{1}\in \mathrm{Z}\text{, }&x\neq 0\text{ and }y\neq 0\text{ and }y\neq 1\\m\in \mathrm{C}\text{, }&x=0\text{ or }y=0\text{ or }\left(y=1\text{ and }x=1\right)\text{ or }\left(y=1\text{ and }x=-1\right)\end{matrix}\right.$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=-y^{\frac{m-1}{2}}\text{; }x=y^{\frac{m-1}{2}}\text{, }&y\neq 0\\x\in \mathrm{C}\text{, }&y=0\text{ and }m\neq 0\end{matrix}\right.$$

Solve for m

$$\left\{\begin{matrix}m=\log_{y}\left(yx^{2}\right)\text{, }&y>0\text{ and }x\neq 0\text{ and }y\neq 1\\m\in \mathrm{R}\text{, }&\left(x=0\text{ and }y>0\right)\text{ or }\left(x=0\text{ and }y<0\text{ and }Denominator(m)\text{bmod}2=1\right)\text{ or }\left(y=1\text{ and }|x|=1\right)\text{ or }\left(y=-1\text{ and }Denominator(m)\text{bmod}2=1\text{ and }Numerator(m)\text{bmod}2=1\text{ and }|x|=1\right)\\m>0\text{, }&y=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=0\text{, }&\left(y<0\text{ and }Denominator(m)\text{bmod}2=1\right)\text{ or }\left(y=0\text{ and }m>0\right)\text{ or }y>0\\x=\sqrt{y^{m-1}}\text{; }x=-\sqrt{y^{m-1}}\text{, }&\left(Denominator(m)\text{bmod}2=1\text{ or }y>0\right)\text{ and }y^{m-1}\geq 0\text{ and }y\neq 0\\x\in \mathrm{R}\text{, }&y=0\text{ and }m>0\end{matrix}\right.$$