#### 1.13 Example 13

$\left ( y^{\prime }\right ) ^{2}-4y=0$ Applying p-discriminant method gives\begin {align*} F & =\left ( y^{\prime }\right ) ^{2}-4y=0\\ \frac {\partial F}{\partial y^{\prime }} & =2y^{\prime }=0 \end {align*}

We ﬁrst check that $$\frac {\partial F}{\partial y}=-4\neq 0$$.  Now we apply p-discriminant.  Eliminating $$y^{\prime }$$. Second equation gives $$y^{\prime }=0$$. Hence ﬁrst equation now gives $$y_{s}=0$$. We see this also satisﬁes the ode. The primitive can be found to be $\Psi \left ( x,y,c\right ) =y-\left ( x+c\right ) ^{2}=0$ Now we have to eliminate $$c$$ using the c-discriminant method\begin {align*} \Psi \left ( x,y,c\right ) & =y-\left ( x+c\right ) ^{2}=0\\ \frac {\partial \Psi \left ( x,y,c\right ) }{\partial c} & =-2\left ( x+c\right ) =0 \end {align*}

Second equation gives $$c=-x$$. Substituting this into the ﬁrst equation gives \begin {align*} y-\left ( x-x\right ) ^{2} & =0\\ y_{s} & =0 \end {align*}

Since 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 $$c$$.