2.7 Example 7 \(y-\left ( y^{\prime }\right ) ^{2}x+\frac {1}{y^{\prime }}=0\)

\begin{align*} y-\left ( y^{\prime }\right ) ^{2}x+\frac {1}{y^{\prime }} & =0\\ y-p^{2}x+\frac {1}{p} & =0 \end{align*}

Applying p-discriminant method gives

\begin{align*} F & =y-p^{2}x+\frac {1}{p}=0\\ \frac {\partial F}{\partial y^{\prime }} & =-2xp-\frac {1}{p^{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 \(p.\)

\begin{align*} p_{1} & =\frac {\left ( -\frac {1}{2}\right ) ^{\frac {1}{3}}}{x^{\frac {1}{3}}}\\ p_{2} & =\frac {1}{2^{\frac {1}{3}}x^{\frac {1}{3}}}\\ p_{3} & =-\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 \(p_{2}=\frac {1}{2^{\frac {1}{3}}x^{\frac {1}{3}}}\). 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 \(p_{3}=-\frac {\left ( -1\right ) ^{\frac {2}{3}}}{2^{\frac {1}{3}}x^{\frac {1}{3}}}\). 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.