e^{y}\mbox{d}x+\left(xe^{y}-2y\right)\mbox{d}y=0\\
e^{y}x^{\prime}+xe^{y}-2y=0\\
e^{y}x^{\prime}+xe^{y}=2y\\
x^{\prime}+x=2ye^{-y}\\
x^{\prime}+x=0\\
x^{\prime}=-x\\
\frac{x^{\prime}}{x}=-1\\
\frac{\mbox{d}x}{x}=-\mbox{d}y\\
\ln{|x|}=-y+C\\
x=Ce^{-y}\\
x\left(y\right)=C\left(y\right)e^{-y}\\
C^{\prime}\left(y\right)e^{-y}-C\left(y\right)e^{-y}+C\left(y\right)e^{-y}=2ye^{-y}\\
C^{\prime}\left(y\right)e^{-y}=2ye^{-y}\\
C^{\prime}\left(y\right)=2y\\
C\left(y\right)=y^2+C\\
x=\left(y^2+C\right)e^{-y}