\[ y-2xy^{\prime }-\sin \left ( y^{\prime }\right ) =0 \] Applying p-discriminant method gives\begin {align*} F & =y-2xy^{\prime }-\sin \left ( y^{\prime }\right ) =0\\ \frac {\partial F}{\partial y^{\prime }} & =-2x-\cos \left ( y^{\prime }\right ) =0 \end {align*}
We first check that \(\frac {\partial F}{\partial y}=1\neq 0\). Now we apply p-discriminant. Second equation gives \(-2x-\cos \left ( y^{\prime }\right ) =0\) or \(y^{\prime }=\arccos \left ( -2x\right ) .\) Substituting in the first equation gives \(y-2x\arccos \left ( -2x\right ) -\sin \left ( \arccos \left ( -2x\right ) \right ) =0.\) I need to look at this more. This should give \(y_{s}=0\) but now it does not.