2.19 Example 19 \(\left ( y^{\prime }\right ) ^{3}-4xyy^{\prime }+8y^{2}=0\)

\begin{align*} \left ( y^{\prime }\right ) ^{3}-4xyy^{\prime }+8y^{2} & =0\\ p^{3}-4xyp+8y^{2} & =0 \end{align*}

Applying p-discriminant method gives

\begin{align*} F & =p^{3}-4xyp+8y^{2}=0\\ \frac {\partial F}{\partial p} & =3p^{2}-4xy=0 \end{align*}

We first check that \(\frac {\partial F}{\partial y}=-4xy^{\prime }+16y\neq 0\).  Now we apply p-discriminant. Eliminating \(p\). Second equation gives \(p=\pm \left ( \frac {4xy}{3}\right ) ^{\frac {1}{2}}\). Substituting first solution in the first equation gives

\begin{align*} \left ( \frac {4xy}{3}\right ) ^{\frac {3}{2}}-4xy\left ( \frac {4xy}{3}\right ) ^{\frac {1}{2}}+8y^{2} & =0\\ y_{s} & =\frac {4}{27}x^{3}\end{align*}

Which satisfies the ode. The general solution can be found to be

\[ \Psi \left ( x,y,c\right ) =y-\frac {1}{4}\frac {x^{2}}{c}+\frac {1}{8}\frac {x}{c^{2}}-\frac {1}{64c^{3}}=0 \]

Hence

\begin{align*} \Psi \left ( x,y,c\right ) & =y-\frac {1}{4}\frac {x^{2}}{c}+\frac {1}{8}\frac {x}{c^{2}}-\frac {1}{64c^{3}}=0\\ \frac {\partial \Psi \left ( x,y,c\right ) }{\partial c} & =\frac {1}{4}\frac {x^{2}}{c^{2}}-\frac {1}{4}\frac {x}{c^{3}}+\frac {3}{64c^{4}}=0 \end{align*}

Eliminating \(c\). Second equation gives \(c=\frac {1}{4x}\) or \(c=\frac {3}{4x}\). Substituting \(c=\frac {1}{4x}\) in the first equation above gives

\begin{align*} y-\frac {1}{4}\frac {x^{2}}{\left ( \frac {1}{4x}\right ) }+\frac {1}{8}\frac {x}{\left ( \frac {1}{4x}\right ) ^{2}}-\frac {1}{64\left ( \frac {1}{4x}\right ) ^{3}} & =0\\ y_{s} & =0 \end{align*}

Which satisfies the ode. But \(y=0\) can be obtained from the general solution above when \(c=\infty \) so it is not singular solution. Substituting \(c=\frac {3}{4x}\) in the first equation above gives

\begin{align*} y-\frac {1}{4}\frac {x^{2}}{\left ( \frac {3}{4x}\right ) }+\frac {1}{8}\frac {x}{\left ( \frac {3}{4x}\right ) ^{2}}-\frac {1}{64\left ( \frac {3}{4x}\right ) ^{3}} & =0\\ y & =\frac {4}{27}x^{3}\end{align*}

Which is the same obtained by p-discriminant. Hence 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 16: Singular solution envelope with general solution curves