Solve 1/3sqrt{x/y}=1/3x*frac{sqrt{xy}}{x}

$$\frac{ 1 }{ 3 } \sqrt{ \frac{ x }{ y } } = \frac{ 1 }{ 3x } \times \frac{ \sqrt{ xy } }{ x }$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=i\sqrt{y}\text{, }&|-\frac{arg(iy^{\frac{3}{2}})}{2}+arg(-\sqrt{iy^{-\frac{1}{2}}}y)|<\pi \text{ and }y\neq 0\\x=-i\sqrt{y}\text{, }&|-\frac{arg(-iy^{\frac{3}{2}})}{2}+arg(-\sqrt{-iy^{-\frac{1}{2}}}y)|<\pi \text{ and }y\neq 0\\x=\sqrt{y}\text{, }&|-\frac{arg(y^{\frac{3}{2}})}{2}+\frac{3arg(y)}{4}|<\pi \text{ and }y\neq 0\\x=-\sqrt{y}\text{, }&|-\frac{arg(-y^{\frac{3}{2}})}{2}+arg(\sqrt{-y^{-\frac{1}{2}}}y)|<\pi \text{ and }y\neq 0\end{matrix}\right.$$

Solve for y (complex solution)

$$\left\{\begin{matrix}y=-x^{2}\text{, }&|-\frac{arg(-x^{3})}{2}+arg(\sqrt{-\frac{1}{x}}x^{2})|<\pi \text{ and }x\neq 0\\y=x^{2}\text{, }&|-\frac{arg(x^{3})}{2}+arg(x^{\frac{3}{2}})|<\pi \text{ and }x\neq 0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\sqrt{y}\text{, }&y>0\\x=-\sqrt{-y}\text{, }&y<0\end{matrix}\right.$$

Solve for y

$$\left\{\begin{matrix}y=-x^{2}\text{, }&x<0\\y=x^{2}\text{, }&x>0\end{matrix}\right.$$