\[ y-x\left ( 1+y^{\prime }\right ) -\left ( y^{\prime }\right ) ^{2}=0 \] Applying p-discriminant method gives\begin {align*} F & =y-x\left ( 1+y^{\prime }\right ) -\left ( y^{\prime }\right ) ^{2}=0\\ \frac {\partial F}{\partial y^{\prime }} & =-x-2y^{\prime }=0 \end {align*}
We first check that \(\frac {\partial F}{\partial y}=1\neq 0\). Now we apply p-discriminant. Second equation gives \(y^{\prime }=-\frac {x}{2}\). Substituting in the first equation gives \(y-x\left ( 1+\left ( -\frac {x}{2}\right ) \right ) -\left ( -\frac {x}{2}\right ) ^{2}=0\) or \(\allowbreak \frac {1}{4}x^{2}-x+y=0\). Hence\[ y_{s}=x-\frac {1}{4}x^{2}\] This does not satisfy the ode. Hence no singular solution exist.