2.4 Example 4 \(y-2xy^{\prime }-\ln y^{\prime }=0\)

\begin{align*} y-2xy^{\prime }-\ln y^{\prime } & =0\\ y-2xp-\ln p & =0 \end{align*}

Applying p-discriminant method via elimination gives

\begin{align*} F & =y-2xp-\ln p=0=0\\ \frac {\partial F}{\partial p} & =-2x-\frac {1}{p}=0 \end{align*}

Second equation gives \(p=-\frac {1}{2x}\). Substituting in the first equation gives

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

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