1.22 Example 22

\[ \left ( y^{\prime }\right ) ^{2}+2xy^{\prime }-y=0 \] Applying p-discriminant method gives\begin {align*} F & =\left ( y^{\prime }\right ) ^{2}+2xy^{\prime }-y=0\\ \frac {\partial F}{\partial y^{\prime }} & =2y^{\prime }+2x=0 \end {align*}

Eliminating \(y^{\prime }\). Second equation gives \(y^{\prime }=-x\). Substituting first solution in the first equation gives \begin {align*} x^{2}-2x^{2}-y & =0\\ y_{s} & =-x^{2} \end {align*}

Now we check that this satisfies the ode itself.  We see it does not. Now we try the c-discriminant method. The general solution is too complicated to write here. But Mathematica and Maple claim there is no singular solution. So will leave it there for now. The paper I took this example from is wrong. It claimed \(y=x^{2}\) is the envelope. It is not.