\[ 27y-8\left ( y^{\prime }\right ) ^{3}=0 \] Applying p-discriminant method gives\begin {align*} F & =27y-8\left ( y^{\prime }\right ) ^{3}=0\\ \frac {\partial F}{\partial y^{\prime }} & =-24\left ( y^{\prime }\right ) ^{2}=0 \end {align*}
We first check that \(\frac {\partial F}{\partial y}=27\neq 0\). Now we apply p-discriminant. Second equation gives \(y^{\prime }=0\). First equation now gives \(27y=0\) or \(y_{s}=0\). We see this also satisfies the ode. The general solution can be found as\[ \Psi \left ( x,y,c\right ) =y^{2}-\left ( x+c\right ) ^{3}=0 \] Applying c-discriminant\begin {align*} \Psi \left ( x,y,c\right ) & =y^{2}-\left ( x+c\right ) ^{3}=0\\ \frac {\partial \Psi \left ( x,y,c\right ) }{\partial c} & =-3\left ( x+c\right ) ^{2}=0 \end {align*}
Second equation gives \(\left ( x+c\right ) ^{2}=0\) or \(c=-x\). From first equation this gives \(y^{2}=0\) or \(y=0\). This is the same as \(y_{s}\) found from p-discriminant, hence\[ y_{s}=0 \] The following plot shows the singular solution as the envelope of the family of general solution plotted using different values of \(c\).