#### 2.31   ODE No. 31

$y'(x)-a x^n \left (y(x)^2+1\right )=0$ Mathematica : cpu = 0.129904 (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 = 0.036 (sec), leaf count = 23

$\left \{ y \left ( x \right ) =\tan \left ( {\frac { \left ( {x}^{n+1}+ \left ( n+1 \right ) {\it \_C1} \right ) a}{n+1}} \right ) \right \}$

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\left ( x\right ) +Q\left ( x\right ) y+R\left ( x\right ) y^{2}\tag {1} \end {align}

This is Ricatti ﬁrst order non-linear ODE. $$P\left ( x\right ) =ax^{n},Q\left ( x\right ) =0,R\left ( x\right ) =ax^{n}$$. 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$\arctan \left ( y\left ( x\right ) \right ) =a\frac {x^{n+1}}{n+1}+C$ Or$y\left ( x\right ) =\tan \left ( a\frac {x^{n+1}}{n+1}+C\right )$

Veriﬁcation

restart;
eq:=diff(y(x),x)-a*x^n*(y(x)^2+1) = 0;
sol:=tan(a*x^(n+1)/(n+1)+_C1);
odetest(y(x)=sol,eq);
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