\[ \left ( y^{\prime }\right ) ^{2}\left ( 2-3y\right ) ^{2}-4\left ( 1-y\right ) =0 \] Applying p-discriminant method gives\begin {align*} F & =\left ( y^{\prime }\right ) ^{2}\left ( 2-3y\right ) ^{2}-4\left ( 1-y\right ) =0\\ \frac {\partial F}{\partial y^{\prime }} & =2y^{\prime }\left ( 2-3y\right ) ^{2}=0 \end {align*}
Eliminating \(y^{\prime }\). Second equation gives \(y^{\prime }=0\) and \(y=\frac {2}{3}\). The solution \(y=\frac {2}{3}\) does not satisfy the ode. Substituting \(y^{\prime }=0\) in the first equation gives\begin {align*} \left ( 1-y\right ) & =0\\ y_{s} & =1 \end {align*}
This solution does satisfy the ode. The following plot shows the singular solution as the envelope of the family of general solution plotted using different values of \(c.\)