\[ y-2xy^{\prime }-\ln y^{\prime }=0 \] Applying p-discriminant method gives\begin {align*} F & =y-2xy^{\prime }-\ln y^{\prime }=0\\ \frac {\partial F}{\partial y^{\prime }} & =-2x-\frac {1}{y^{\prime }}=0 \end {align*}
Second equation gives \(y^{\prime }=-\frac {1}{2x}\). Substituting in the first equation gives \begin {align*} y+1-\ln \frac {-1}{2x} & =0\\ y_{s} & =\ln \left ( \frac {-1}{2x}\right ) -1 \end {align*}
This does not satisfy the ode. Hence no singular solution exist.