Link to actual problem [9316] \[ \boxed {y^{\prime }-\frac {y^{3}-3 y^{2} x +3 y x^{2}-x^{3}+x^{2}}{\left (x -1\right ) \left (1+x \right )}=0} \]
type detected by program
{"abelFirstKind", "first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Abel]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\).\begin{align*} \\ \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= -\frac {\ln \left (x^{2}-2 x y+y^{2}+x -y+1\right )}{6}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 y-2 x -1\right ) \sqrt {3}}{3}\right )}{3}+\frac {\ln \left (y-x +1\right )}{3}\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 &=x^{3}-3 x^{2} y +3 x \,y^{2}-y^{3}-1 \\ \frac {dS}{dR} &= -\frac {1}{R^{2}-1} \\ \end{align*}