#### 1.19 Example 19

$\left ( y^{\prime }\right ) ^{3}-4xyy^{\prime }+8y^{2}=0$ Applying p-discriminant method gives\begin {align*} F & =\left ( y^{\prime }\right ) ^{3}-4xyy^{\prime }+8y^{2}=0\\ \frac {\partial F}{\partial y^{\prime }} & =3\left ( y^{\prime }\right ) ^{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 $$y^{\prime }$$. Second equation gives $$y^{\prime }=\pm \left ( \frac {4xy}{3}\right ) ^{\frac {1}{2}}$$. 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 $$c$$.