\[
\boxed{
{\frac {\rm d}{{\rm d}x}}y \left( x \right) ={\frac {-\ln  \left( x \right) +2\,\ln  \left( 2\,x \right) xy \left( x \right) +\ln  \left( 2\,x \right) +\ln  \left( 2\,x \right)  \left( y \left( x \right)  \right) ^{2}+\ln  \left( 2\,x \right) {x}^{2}}{\ln  \left( x \right) }}=0}
\]