2.13 Example 13 \(\left ( y^{\prime }\right ) ^{2}-4y=0\)

\begin{align*} \left ( y^{\prime }\right ) ^{2}-4y & =0\\ F & =p^{2}-4y\\ & =0 \end{align*}

This is quadratic in \(p\).

\begin{align*} b^{2}-4ac & =0\\ \left ( 0\right ) -4\left ( 1\right ) \left ( -4y\right ) & =0\\ y & =0 \end{align*}

We see this also satisfies the ode. Hence it is the envelope. The primitive can be found to be

\begin{align*} \Psi \left ( x,y,c\right ) & =y-\left ( x+c\right ) ^{2}=0\\ & =y-\left ( x^{2}+c^{2}+2xc\right ) =0\\ & =y-x^{2}-c^{2}-2xc=0 \end{align*}

This is quadratic in \(c\).

\begin{align*} b^{2}-4aC & =0\\ \left ( -2x\right ) ^{2}-4\left ( -1\right ) \left ( y-x^{2}\right ) & =0\\ 4x^{2}+4y-4x^{2} & =0\\ y & =0 \end{align*}

This is the same as found by p-discriminant method then this is the singular solution. The following plot shows the singular solution as the envelope of the family of general solution plotted using different values of \(c\).

Figure 10: Singular solution envelope with general solution curves