\[ \left ( y^{\prime }\right ) ^{2}=xy \] Applying p-discriminant method gives\begin {align*} F & =\left ( y^{\prime }\right ) ^{2}-xy=0\\ \frac {\partial F}{\partial y^{\prime }} & =2y^{\prime }=0 \end {align*}
We first check that \(\frac {\partial F}{\partial y}=-x\neq 0\). Now we apply p-discriminant. Second equation gives \(y^{\prime }=0\). Hence first equation now gives \(xy=0\) or \(y_{s}=0\). We see this satisfies the ode. Now we have to find the general solution. It will be\begin {align*} y & =\frac {1}{36}\left ( 4x^{3}-12x^{\frac {3}{2}}c+9c^{2}\right ) \\ y & =\frac {1}{36}\left ( 4x^{3}+12x^{\frac {3}{2}}c+9c^{2}\right ) \end {align*}
Hence we have two general solutions. These can be written as\begin {align*} \Psi _{1}\left ( x,y,c\right ) & =y-\frac {1}{36}\left ( 4x^{3}-12x^{\frac {3}{2}}c+9c^{2}\right ) =0\\ \Psi _{2}\left ( x,y,c\right ) & =y-\frac {1}{36}\left ( 4x^{3}+12x^{\frac {3}{2}}c+9c^{2}\right ) =0 \end {align*}
Now we have to eliminate \(c\) from each and see if the resulting \(y\) solution agrees with the one found from the one found from the p-discriminant method. Starting with the first one\begin {align*} \Psi _{1}\left ( x,y,c\right ) & =y-\frac {1}{36}\left ( 4x^{3}-12x^{\frac {3}{2}}c+9c^{2}\right ) =0\\ \frac {\partial \Psi _{1}\left ( x,y,c\right ) }{\partial c} & =-\frac {1}{36}\left ( -12x^{\frac {3}{2}}+18c\right ) =0 \end {align*}
Second equation gives \(c=\frac {12}{18}x^{\frac {3}{2}}=\frac {2}{3}x^{\frac {3}{2}}\). Substituting this in the first equation gives\begin {align*} y-\frac {1}{36}\left ( 4x^{3}-12x^{\frac {3}{2}}\left ( \frac {2}{3}x^{\frac {3}{2}}\right ) +9\left ( \frac {2}{3}x^{\frac {3}{2}}\right ) ^{2}\right ) & =0\\ y & =0 \end {align*}
Which agrees with \(y_{s}=0\) found from the p-discriminant method. For the second general solution\begin {align*} \Psi _{2}\left ( x,y,c\right ) & =y-\frac {1}{36}\left ( 4x^{3}+12x^{\frac {3}{2}}c+9c^{2}\right ) =0\\ \frac {\partial \Psi _{2}\left ( x,y,c\right ) }{\partial c} & =-\frac {1}{36}\left ( 12x^{\frac {3}{2}}+18c\right ) =0 \end {align*}
Second equation gives \(c=-\frac {12}{18}x^{\frac {3}{2}}=-\frac {2}{3}x^{\frac {3}{2}}\). Substituting this in the first equation gives\begin {align*} y-\frac {1}{36}\left ( 4x^{3}+12x^{\frac {3}{2}}\left ( -\frac {2}{3}x^{\frac {3}{2}}\right ) +9\left ( -\frac {2}{3}x^{\frac {3}{2}}\right ) ^{2}\right ) & =0\\ y & =0 \end {align*}
Which agrees with \(y_{s}=0\) found from the p-discriminant method. Hence \(y_{s}=0\) is singular solution. The following plot shows the singular solution as the envelope of the family of general solution plotted using different values of \(c\).