1.3.1.2.2 Example  \(y^{\prime }=x^{-4}+y^{2}\)

\[ y^{\prime }=x^{-4}+y^{2}\]

Comparing this to \(y^{\prime }=ax^{n}+by^{2}\) shows that \(a=1,b=1,n=-4\). We see that \(n\) satisfies that \(\frac {n}{2n+4}=1\) which is integer. Hence we expect that applying (2) will give solution in elementary functions. Since \(ab>0\) then applying

\begin{align*} w & =\sqrt {x}c_{1}\operatorname {BesselJ}\left ( \frac {1}{2k},\frac {1}{k}\sqrt {ab}x^{k}\right ) +c_{2}\operatorname {BesselY}\left ( \frac {1}{2k},\frac {1}{k}\sqrt {ab}x^{k}\right ) \\ k & =1+\frac {-4}{2}=1-2=-1 \end{align*}

Hence

\[ w=\sqrt {x}c_{1}\operatorname {BesselJ}\left ( \frac {-1}{2},-x^{-1}\right ) +c_{2}\operatorname {BesselY}\left ( -\frac {1}{2},-x^{-1}\right ) \]

Hence

\[ y=-\frac {w^{\prime }}{w}\]

Simplifying the above gives

\[ y=\frac {1}{x^{2}}\left ( \tan \left ( -\frac {1}{x}+c_{1}\right ) -x\right ) \]