2.5 Example 5 \(y-x\left ( 1+y^{\prime }\right ) -\left ( y^{\prime }\right ) ^{2}=0\)

\begin{align*} y-x\left ( 1+y^{\prime }\right ) -\left ( y^{\prime }\right ) ^{2} & =0\\ F & =y-x-xp-p^{2}\\ & =0 \end{align*}

We first check that \(\frac {\partial F}{\partial y}=1\neq 0\). Since this is quadratic in \(p\), we do not have to use elimination and can just use the quadratic discriminant

\begin{align*} b^{2}-4ac & =0\\ \left ( -x\right ) ^{2}-4\left ( -1\right ) \left ( y-x\right ) & =0\\ x^{2}+4y-4x & =0\\ y & =x-\frac {1}{4}x^{2}\end{align*}

This does not satisfy the ode. Hence no singular solution exist.