Link to actual problem [9271] \[ \boxed {y^{\prime }-\frac {-x^{2}+x +1+y^{2}+5 x^{2} y-2 x y+4 x^{4}-3 x^{3}+y^{3}+3 y^{2} x^{2}-3 y^{2} x +3 y x^{4}-6 x^{3} y+x^{6}-3 x^{5}}{x}=0} \]
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*} \\ \left [R &= x, S \left (R \right ) &= \int _{}^{y}\frac {1}{x^{6}-3 x^{5}+\left (3 \textit {\_a} +4\right ) x^{4}+\left (-6 \textit {\_a} -3\right ) x^{3}+\left (3 \textit {\_a}^{2}+5 \textit {\_a} +1\right ) x^{2}+\left (-3 \textit {\_a}^{2}-2 \textit {\_a} \right ) x +\textit {\_a}^{3}+\textit {\_a}^{2}+1}d \textit {\_a}\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 &= -\frac {x}{2} \\ \eta &=x^{2}-\frac {1}{2} x \\ \frac {dS}{dR} &= -\frac {2}{R^{3}+R^{2}+1} \\ \end{align*}