2.31   ODE No. 31

\[ y'(x)-a x^n \left (y(x)^2+1\right )=0 \] Mathematica : cpu = 0.178547 (sec), leaf count = 21


\[\left \{\left \{y(x)\to \tan \left (\frac {a x^{n+1}}{n+1}+c_1\right )\right \}\right \}\] Maple : cpu = 0.058 (sec), leaf count = 23


\[y \relax (x ) = \tan \left (\frac {a \left (x^{n +1}+\left (n +1\right ) c_{1}\right )}{n +1}\right )\]

Hand solution

\begin {align} y^{\prime }-ax^{n}\left (y^{2}+1\right ) & =0\nonumber \\ y^{\prime } & =ax^{n}+ax^{n}y^{2}\nonumber \\ & =P\relax (x) +Q\relax (x) y+R\relax (x) y^{2}\tag {1} \end {align}

This is Ricatti first order non-linear ODE. \(P\relax (x) =ax^{n},Q\relax (x) =0,R\relax (x) =ax^{n}\). But this is separable also. Hence\begin {align*} \frac {y^{\prime }}{\left (y^{2}+1\right ) } & =ax^{n}\\ \frac {dy}{\left (y^{2}+1\right ) } & =ax^{n}dx \end {align*}

Integrating\[ \arctan \left (y\relax (x) \right ) =a\frac {x^{n+1}}{n+1}+C \] Or\[ y\relax (x) =\tan \left (a\frac {x^{n+1}}{n+1}+C\right ) \]

Verification