y^{\prime}=\frac{y}{x}\left(1+\ln{y}-\ln{x}\right)\\
y^{\prime}=\frac{y}{x}\left(1+\ln{\frac{y}{x}}\right)\\
y=ux\\
y^{\prime}=u^{\prime}x+u\\
u^{\prime}x+u=u\left(1+\ln{u}\right)\\
u^{\prime}x=u\ln{u}\\
\frac{u^{\prime}}{u\ln{u}}=\frac{1}{x}\\
\frac{\mbox{d}u}{u\ln{u}}=\frac{\mbox{d}x}{x}\\
\ln{\ln{u}}=\ln{x}+C\\
\ln{\ln{u}}=\ln{Cx}\\
\ln{u}=Cx\\
\ln{y}-\ln{x}=Cx\\
\ln{y}=Cx+\ln{x}\\
y=xe^{Cx}