2.20 Example 20 \(4x\left ( y^{\prime }\right ) ^{2}=\left ( 2x-1\right ) ^{2}\)

\begin{align*} 4x\left ( y^{\prime }\right ) ^{2} & =\left ( 2x-1\right ) ^{2}\\ 4xp^{2} & =\left ( 2x-1\right ) ^{2}\\ F & =-4xp^{2}+\left ( 2x-1\right ) ^{2}\\ & =0 \end{align*}

We first check that \(\frac {\partial F}{\partial y}\) and see it is zero. Hence no singular solution exists.