Example 8 \begin {equation} \left ( 1+\left ( y^{\prime }\right ) ^{2}\right ) ^{2}=y^{2}y^{\prime \prime } \tag {1} \end {equation} With IC\begin {align*} y\left ( 0\right ) & =3\\ y^{\prime }\left ( 0\right ) & =\sqrt {2} \end {align*} Let \(p=y^{\prime }\), hence \(y^{\prime \prime }=pp^{\prime }\) and the ode becomes\begin {align} \left ( 1+p^{2}\right ) ^{2} & =y^{2}pp^{\prime }\nonumber \\ \frac {pp^{\prime }}{\left ( 1+p^{2}\right ) ^{2}} & =\frac {1}{y^{2}} \tag {2} \end {align}

Solving the above ode gives\begin {align} p_{1} & =-\frac {\sqrt {-2\left ( c_{1}y-1\right ) \left ( 2c_{1}y+y-2\right ) }}{2\left ( c_{1}y-1\right ) }\tag {3}\\ p_{2} & =\frac {\sqrt {-2\left ( c_{1}y-1\right ) \left ( 2c_{1}y+y-2\right ) }}{2\left ( c_{1}y-1\right ) } \tag {4} \end {align}

Now we replace back \(p=y^{\prime }\left ( x\right ) \) above gives\begin {align} y^{\prime } & =-\frac {\sqrt {-2\left ( c_{1}y-1\right ) \left ( 2c_{1}y+y-2\right ) }}{2\left ( c_{1}y-1\right ) }\tag {3A}\\ y^{\prime } & =\frac {\sqrt {-2\left ( c_{1}y-1\right ) \left ( 2c_{1}y+y-2\right ) }}{2\left ( c_{1}y-1\right ) } \tag {4A} \end {align}

Lets start with (3A). Before solving, we will ge rid of \(c_{1}\) using IC. Given that \(y\left ( 0\right ) =3,y^{\prime }\left ( 0\right ) =\sqrt {2}\) then (3A) becomes\begin {align*} \sqrt {2} & =-\frac {\sqrt {-2\left ( 3c_{1}-1\right ) \left ( 6c_{1}+3-2\right ) }}{2\left ( 3c_{1}-1\right ) }\\ c_{1} & =\frac {1}{6} \end {align*}

Hence (4A) becomes\[ y^{\prime }=-\frac {\sqrt {-2\left ( \frac {1}{6}y-1\right ) \left ( 3y+y-2\right ) }}{2\left ( \frac {1}{6}y-1\right ) }\] Solving this ode gives the solution \begin {equation} x-\frac {\sqrt {-4y^{2}+30y-36}}{4}-\frac {9\arcsin \left ( \frac {4y}{9}-\frac {5}{3}\right ) }{8}+c_{2}=0 \tag {5} \end {equation} Finally, using \(y\left ( 0\right ) =3\) the above becomes\[ -\frac {\sqrt {-4\left ( 9\right ) +30\left ( 3\right ) -36}}{4}-\frac {9\arcsin \left ( \frac {4\left ( 3\right ) }{9}-\frac {5}{3}\right ) }{8}+c_{2}=0 \] Solving for \(c_{2}\) gives\[ c_{2}=\frac {\sqrt {18}}{4}-\frac {9\arcsin \left ( \frac {1}{3}\right ) }{8}\] Hence (5) becomes\begin {equation} x-\frac {\sqrt {-4y^{2}+30y-36}}{4}-\frac {9\arcsin \left ( \frac {4y}{9}-\frac {5}{3}\right ) }{8}+\frac {\sqrt {18}}{4}-\frac {9\arcsin \left ( \frac {1}{3}\right ) }{8}=0 \tag {6} \end {equation} Now we have to do the same for ode (4A).  Given that \(y\left ( 0\right ) =3,y^{\prime }\left ( 0\right ) =\sqrt {2}\) then (4A) becomes\[ \sqrt {2}=\frac {\sqrt {-2\left ( 3c_{1}-1\right ) \left ( 6c_{1}+3-2\right ) }}{2\left ( 3c_{1}-1\right ) }\] But there is no solution for \(c_{1}\). This means (4A) leads to no solution. Hence only solution is (6).