2\mbox{d}x+\sqrt{\frac{x}{y}}\mbox{d}y-\sqrt{\frac{y}{x}}\mbox{d}x=0\\
\left(2-\sqrt{\frac{y}{x}}\right)\mbox{d}x+\sqrt{\frac{x}{y}}\mbox{d}y=0\\
\left(2-\sqrt{\frac{y}{x}}\right)=-\sqrt{\frac{x}{y}}y^{\prime}\\
y=ux\\
y^{\prime}=u^{\prime}x+u\\
\left(2-\sqrt{u}\right)=-\frac{1}{\sqrt{u}}\left(u^{\prime}x+u\right)\\
u-2\sqrt{u}=u^{\prime}x+u\\
-2\sqrt{u}=u^{\prime}x\\
\frac{u^{\prime}}{2\sqrt{u}}=-\frac{1}{x}\\
\frac{\mbox{d}u}{2\sqrt{u}}=-\frac{\mbox{d}x}{x}\\
\sqrt{u}=-\ln{|x|}+C\\
u=\left(-\ln{|x|}+C\right)^2\\
\frac{y}{x}=\left(-\ln{|x|}+C\right)^2\\
y=x\left(-\ln{|x|}+C\right)^2