2.15 \(y-\left ( y^{\prime }\right ) ^{2}+3xy^{\prime }-3x^{2}=0\) \begin{align*} y-\left ( y^{\prime }\right ) ^{2}+3xy^{\prime }-3x^{2} & =0\\ y-p^{2}+3xp-3x^{2} & =0\\ F & =y-p^{2}+3xp-3x^{2}\\ & =0 \end{align*}
We first check that \(\frac {\partial F}{\partial y}=1\neq 0\) . This is quadratic in \(p\) .
\begin{align*} b^{2}-4ac & =0\\ \left ( 3x\right ) ^{2}-4\left ( -1\right ) \left ( y-3x^{2}\right ) & =0\\ 9x^{2}+4y-12x^{2} & =0\\ y & =\frac {3}{4}x^{2}\end{align*}
Which satisfies the ode. Hence it is the envelope. The primitive can be found to be
\[ \Psi \left ( x,y,c\right ) =y-cx-c^{2}-x^{2}=0 \]
This is
quadratic in \(c\) .
\begin{align*} b^{2}-4aC & =0\\ \left ( -x\right ) ^{2}-4\left ( -1\right ) \left ( y-x^{2}\right ) & =0\\ x^{2}+4y-4x^{2} & =0\\ y & =\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 \(c\) .