Link to actual problem [9221] \[ \boxed {y^{\prime }-\frac {6 x^{2} y-2 x +1-5 x^{3} y^{2}-2 x y+y^{3} x^{4}}{x^{2} \left (x^{2} y-x +1\right )}=0} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[_rational, [_Abel, `2nd type`, `class C`], [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\).\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (x y -1\right ) \left (2 x^{3} y^{2}-6 x^{2} y +4 x -1\right )}{2 x \left (x^{2} y -x +1\right )}\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^{4} y^{3}-3 x^{3} y^{2}+3 x^{2} y -x}{2 x^{2} y -2 x +2} \\ \frac {dS}{dR} &= \frac {2}{R^{2}} \\ \end{align*}