Link to actual problem [9070] \[ \boxed {y^{\prime }-\frac {2 x^{2}+2 x +x^{4}-2 x^{2} y-1+y^{2}}{x +1}=0} \]
type detected by program
{"riccati", "first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\).\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {\ln \left (-x^{2}+y-1\right )}{2}-\frac {\ln \left (-x^{2}+y+1\right )}{2}\right ] \\ \end{align*}
My program’s symgen result This shows my program’s found \(\xi ,\eta \) and the corresponding ODE in canonical coordinates \(R,S\).\begin{align*} \xi &= 0 \\ \eta &=\frac {-x^{5}-x^{4}+2 x^{3} y +2 x^{2} y -x \,y^{2}-y^{2}+x +1}{2 x +2} \\ \frac {dS}{dR} &= -\frac {2}{R +1} \\ \end{align*}