#### 1.3 Example 3

$27y-8\left ( y^{\prime }\right ) ^{3}=0$ Applying p-discriminant method gives\begin {align*} F & =27y-8\left ( y^{\prime }\right ) ^{3}=0\\ \frac {\partial F}{\partial y^{\prime }} & =-24\left ( y^{\prime }\right ) ^{2}=0 \end {align*}

We ﬁrst check that $$\frac {\partial F}{\partial y}=27\neq 0$$.  Now we apply p-discriminant.  Second equation gives $$y^{\prime }=0$$. First equation now gives $$27y=0$$ or $$y_{s}=0$$. We see this also satisﬁes the ode. The general solution can be found as$\Psi \left ( x,y,c\right ) =y^{2}-\left ( x+c\right ) ^{3}=0$ Applying c-discriminant\begin {align*} \Psi \left ( x,y,c\right ) & =y^{2}-\left ( x+c\right ) ^{3}=0\\ \frac {\partial \Psi \left ( x,y,c\right ) }{\partial c} & =-3\left ( x+c\right ) ^{2}=0 \end {align*}

Second equation gives $$\left ( x+c\right ) ^{2}=0$$ or $$c=-x$$. From ﬁrst equation this gives $$y^{2}=0$$ or $$y=0$$. This is the same as $$y_{s}$$ found from p-discriminant, hence$y_{s}=0$ The following plot shows the singular solution as the envelope of the family of general solution plotted using diﬀerent values of $$c$$.