#### 1.7 Example 7

$y-\left ( y^{\prime }\right ) ^{2}x+\frac {1}{y^{\prime }}=0$ Applying p-discriminant method gives\begin {align*} F & =y-\left ( y^{\prime }\right ) ^{2}x+\frac {1}{y^{\prime }}=0\\ \frac {\partial F}{\partial y^{\prime }} & =-2xy^{\prime }-\frac {1}{\left ( y^{\prime }\right ) ^{2}}=0 \end {align*}

We ﬁrst check that $$\frac {\partial F}{\partial y}=1\neq 0$$.  Now we apply p-discriminant.  Second equation gives 3 solutions for $$y^{\prime }.$$\begin {align*} y^{\prime } & =\frac {\left ( -\frac {1}{2}\right ) ^{\frac {1}{3}}}{x^{\frac {1}{3}}}\\ y^{\prime } & =\frac {1}{2^{\frac {1}{3}}x^{\frac {1}{3}}}\\ y^{\prime } & =-\frac {\left ( -1\right ) ^{\frac {2}{3}}}{2^{\frac {1}{3}}x^{\frac {1}{3}}} \end {align*}

Using the ﬁrst solution, then the ﬁrst equation gives\begin {align*} y-\left ( \frac {\left ( -\frac {1}{2}\right ) ^{\frac {1}{3}}}{x^{\frac {1}{3}}}\right ) ^{2}x+\frac {1}{\left ( \frac {\left ( -\frac {1}{2}\right ) ^{\frac {1}{3}}}{x^{\frac {1}{3}}}\right ) } & =0\\ y_{s} & =\frac {3}{2}\left ( -1\right ) ^{\frac {2}{3}}\sqrt [3]{2}\sqrt [3]{x} \end {align*}

Now we check if this satisﬁes the ode $$F=0$$. It does not. Trying the second solution $$y^{\prime }=\frac {1}{2^{\frac {1}{3}}x^{\frac {1}{3}}}$$. Substituting into $$F=0$$ gives\begin {align*} y-\left ( \frac {1}{2^{\frac {1}{3}}x^{\frac {1}{3}}}\right ) ^{2}x+\frac {1}{\left ( \frac {1}{2^{\frac {1}{3}}x^{\frac {1}{3}}}\right ) } & =0\\ y & =-\frac {1}{2}\sqrt [3]{2}\sqrt [3]{x} \end {align*}

Now we check if this satisﬁes the ode $$F=0$$. It does not. Trying the third solution $$y^{\prime }=-\frac {\left ( -1\right ) ^{\frac {2}{3}}}{2^{\frac {1}{3}}x^{\frac {1}{3}}}$$. Substituting into $$F=0$$ gives\begin {align*} y-\left ( -\frac {\left ( -1\right ) ^{\frac {2}{3}}}{2^{\frac {1}{3}}x^{\frac {1}{3}}}\right ) ^{2}x+\frac {1}{\left ( -\frac {\left ( -1\right ) ^{\frac {2}{3}}}{2^{\frac {1}{3}}x^{\frac {1}{3}}}\right ) } & =0\\ y & =-\frac {3}{2}\sqrt [3]{-1}\sqrt [3]{2}\sqrt [3]{x} \end {align*}

Now we check if this satisﬁes the ode $$F=0$$. It does not. Hence no singular exist.