2.6 Example 6 \(y-2xy^{\prime }-\sin \left ( y^{\prime }\right ) =0\)

\begin{align*} y-2xy^{\prime }-\sin \left ( y^{\prime }\right ) & =0\\ y-2xp-\sin \left ( p\right ) & =0 \end{align*}

Applying p-discriminant method gives

\begin{align*} F & =y-2xp-\sin \left ( p\right ) =0\\ \frac {\partial F}{\partial y^{\prime }} & =-2x-\cos \left ( p\right ) =0 \end{align*}

We first check that \(\frac {\partial F}{\partial y}=1\neq 0\).  Now we apply p-discriminant.  Second equation gives \(-2x-\cos \left ( p\right ) =0\) or \(p=\arccos \left ( -2x\right ) .\) 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.