\[ 4x\left ( y^{\prime }\right ) ^{2}=\left ( 2x-1\right ) ^{2}\] Applying p-discriminant method gives\begin {align*} F & =4x\left ( y^{\prime }\right ) ^{2}-\left ( 2x-1\right ) ^{2}=0\\ \frac {\partial F}{\partial y^{\prime }} & =8xy^{\prime }=0 \end {align*}
We first check that \(\frac {\partial F}{\partial y}\) and see it is zero. Hence no singular solution exists.