1.1 Example 1

$9\left ( y^{\prime }\right ) ^{2}\left ( 2-y\right ) ^{2}=4\left ( 3-y\right )$ Hence \begin {align*} F & =9\left ( y^{\prime }\right ) ^{2}\left ( 2-y\right ) ^{2}-4\left ( 3-y\right ) =0\\ \frac {\partial F}{\partial y^{\prime }} & =18y^{\prime }\left ( 2-y\right ) ^{2}=0 \end {align*}

We ﬁrst check that $$\frac {\partial F}{\partial y}=-18\left ( y^{\prime }\right ) ^{2}\left ( 2-y\right ) +4\neq 0$$.  Now we apply p-discriminant. Eliminating $$y^{\prime }$$. Second equation gives $$y^{\prime }=0$$. Substituting into the ﬁrst equation gives these candidate singular solutions\begin {equation} y_{s}=3 \tag {1} \end {equation} We now have to check if this solution satisﬁes the ode. We see it does.

Now we have to ﬁnd the general solution (also called the primitive). This comes out to be$\Psi \left ( x,y,c\right ) =\left ( x+c\right ) ^{2}-y^{2}\left ( 3-y\right )$ Now we set up\begin {align*} \Psi \left ( x,y,c\right ) & =0=\left ( x+c\right ) ^{2}-y^{2}\left ( 3-y\right ) \\ \frac {\partial \Psi }{\partial c} & =0=2\left ( x+c\right ) \end {align*}

Eliminating $$c$$. Second equation gives $$c=-x$$. Substituting into the ﬁrst equation gives$0=-y^{2}\left ( 3-y\right )$ Hence the c-discriminant method gives\begin {align} y_{s} & =0\tag {2}\\ y_{s} & =3\nonumber \end {align}

Now we take the common $$y_{s}$$ from the p-discriminant and the c-discriminant from (1,2). We see that $$y_{s}=3$$ is common. Hence$y_{s}=3$ And $$y_{s}=0$$ is removed. We also see that $$y_{s}=0$$ does not even satisfy the ode. But even if it did, it is removed since it is not common with the p-discriminant .

If there is no common $$y_{s}$$ found from applying the two method (p-discriminant and the c-discriminant) then it means there is no singular solution. The following plot shows the singular solution as the envelope of the family of general solution plotted using diﬀerent values of $$c$$.