\[ \left ( y^{\prime }\right ) ^{2}-4y=0 \] Applying p-discriminant method gives\begin {align*} F & =\left ( y^{\prime }\right ) ^{2}-4y=0\\ \frac {\partial F}{\partial y^{\prime }} & =2y^{\prime }=0 \end {align*}
We first check that \(\frac {\partial F}{\partial y}=-4\neq 0\). Now we apply p-discriminant. Eliminating \(y^{\prime }\). Second equation gives \(y^{\prime }=0\). Hence first equation now gives \(y_{s}=0\). We see this also satisfies the ode. The primitive can be found to be \[ \Psi \left ( x,y,c\right ) =y-\left ( x+c\right ) ^{2}=0 \] Now we have to eliminate \(c\) using the c-discriminant method\begin {align*} \Psi \left ( x,y,c\right ) & =y-\left ( x+c\right ) ^{2}=0\\ \frac {\partial \Psi \left ( x,y,c\right ) }{\partial c} & =-2\left ( x+c\right ) =0 \end {align*}
Second equation gives \(c=-x\). Substituting this into the first equation gives \begin {align*} y-\left ( x-x\right ) ^{2} & =0\\ y_{s} & =0 \end {align*}
Since this is the same as found by p-discriminant method then this is the singular solution. The following plot shows the singular solution as the envelope of the family of general solution plotted using different values of \(c\).