1.15 Example 15

\[ y-\left ( y^{\prime }\right ) ^{2}+3xy^{\prime }-3x^{2}=0 \] Applying p-discriminant method gives\begin {align*} F & =y-\left ( y^{\prime }\right ) ^{2}+3xy^{\prime }-3x^{2}=0\\ \frac {\partial F}{\partial y^{\prime }} & =-2y^{\prime }+3x=0 \end {align*}

We first check that \(\frac {\partial F}{\partial y}=1\neq 0\).  Now we apply p-discriminant.  Eliminating \(y^{\prime }\). Second equation gives \(y^{\prime }=\frac {3}{2}x\). Substituting into the first equation gives\begin {align*} y-\left ( \frac {3}{2}x\right ) ^{2}+3x\left ( \frac {3}{2}x\right ) -3x^{2} & =0\\ y_{s} & =\frac {3}{4}x^{2} \end {align*}

The primitive can be found to be \[ \Psi \left ( x,y,c\right ) =y-cx-c^{2}-x^{2}=0 \] Now we have to eliminate \(c\) using the c-discriminant method\begin {align*} \Psi \left ( x,y,c\right ) & =y-cx-c^{2}-x^{2}=0\\ \frac {\partial \Psi \left ( x,y,c\right ) }{\partial c} & =-x-2c=0 \end {align*}

Second equation gives \(c=-\frac {x}{2}\). Substituting this into the first equation gives \begin {align*} y-\left ( -\frac {x}{2}\right ) x-\left ( -\frac {x}{2}\right ) ^{2}-x^{2} & =0\\ y_{s} & =\frac {3}{4}x^{2} \end {align*}

Which agrees with the p-discriminant curve. Hence this is a singular solution. The following plot shows the singular solution as the envelope of the family of general solution plotted using different values of \(c\).