2.31   ODE No. 31

1. Problem in Latex
2. Mathematica input
3. Maple input

$$y^{\prime }(x)-a{x}^n(y(x{)}^2+1)=0$$ ✓ Mathematica : cpu = 0.13142 (sec), leaf count = 21

$$\left\{\{y(x)\to \tan (\frac{a{x}^{n+1}}{n+1}+c_1)\}\right\}$$

✓ Maple : cpu = 0.06 (sec), leaf count = 23

$$\{y\left(x\right)=\tan \left(\frac{(x^{n+1}+(n+1)\mathit{\_}\mathit{C}\mathit{1})a}{n+1}\right)\}$$

Hand solution

$$\begin{array}{rl}{y}^{\prime }-a{x}^n(y^2+1)& =0\\ y^{\prime }& =a{x}^n+a{x}^n{y}^2\\ \text{(1)}& & =P\left(x\right)+Q\left(x\right)y+R\left(x\right)y^2\end{array}$$

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

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

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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