#### 1.2 Example 2

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

We ﬁrst check that $$\frac {\partial F}{\partial y}=-x\neq 0$$.  Now we apply p-discriminant.  Second equation gives $$y^{\prime }=0$$. Hence ﬁrst equation now gives $$xy=0$$ or $$y_{s}=0$$. We see this satisﬁes the ode. Now we have to ﬁnd the general solution. It will be\begin {align*} y & =\frac {1}{36}\left ( 4x^{3}-12x^{\frac {3}{2}}c+9c^{2}\right ) \\ y & =\frac {1}{36}\left ( 4x^{3}+12x^{\frac {3}{2}}c+9c^{2}\right ) \end {align*}

Hence we have two general solutions. These can be written as\begin {align*} \Psi _{1}\left ( x,y,c\right ) & =y-\frac {1}{36}\left ( 4x^{3}-12x^{\frac {3}{2}}c+9c^{2}\right ) =0\\ \Psi _{2}\left ( x,y,c\right ) & =y-\frac {1}{36}\left ( 4x^{3}+12x^{\frac {3}{2}}c+9c^{2}\right ) =0 \end {align*}

Now we have to eliminate $$c$$ from each and see if the resulting $$y$$ solution agrees with the one found from the one found from the p-discriminant method. Starting with the ﬁrst one\begin {align*} \Psi _{1}\left ( x,y,c\right ) & =y-\frac {1}{36}\left ( 4x^{3}-12x^{\frac {3}{2}}c+9c^{2}\right ) =0\\ \frac {\partial \Psi _{1}\left ( x,y,c\right ) }{\partial c} & =-\frac {1}{36}\left ( -12x^{\frac {3}{2}}+18c\right ) =0 \end {align*}

Second equation gives $$c=\frac {12}{18}x^{\frac {3}{2}}=\frac {2}{3}x^{\frac {3}{2}}$$. Substituting this in the ﬁrst equation gives\begin {align*} y-\frac {1}{36}\left ( 4x^{3}-12x^{\frac {3}{2}}\left ( \frac {2}{3}x^{\frac {3}{2}}\right ) +9\left ( \frac {2}{3}x^{\frac {3}{2}}\right ) ^{2}\right ) & =0\\ y & =0 \end {align*}

Which agrees with $$y_{s}=0$$ found from the p-discriminant method. For the second general solution\begin {align*} \Psi _{2}\left ( x,y,c\right ) & =y-\frac {1}{36}\left ( 4x^{3}+12x^{\frac {3}{2}}c+9c^{2}\right ) =0\\ \frac {\partial \Psi _{2}\left ( x,y,c\right ) }{\partial c} & =-\frac {1}{36}\left ( 12x^{\frac {3}{2}}+18c\right ) =0 \end {align*}

Second equation gives $$c=-\frac {12}{18}x^{\frac {3}{2}}=-\frac {2}{3}x^{\frac {3}{2}}$$. Substituting this in the ﬁrst equation gives\begin {align*} y-\frac {1}{36}\left ( 4x^{3}+12x^{\frac {3}{2}}\left ( -\frac {2}{3}x^{\frac {3}{2}}\right ) +9\left ( -\frac {2}{3}x^{\frac {3}{2}}\right ) ^{2}\right ) & =0\\ y & =0 \end {align*}

Which agrees with $$y_{s}=0$$ found from the p-discriminant method. Hence $$y_{s}=0$$ is singular solution. The following plot shows the singular solution as the envelope of the family of general solution plotted using diﬀerent values of $$c$$.