Link to actual problem [9282] \[ \boxed {y^{\prime }-\frac {x^{2} y+x^{4}+2 x^{3}-3 x^{2}+x y+x +y^{3}+3 y^{2} x^{2}-3 y^{2} x +3 y x^{4}-6 x^{3} y+x^{6}-3 x^{5}}{x \left (y+x^{2}-x +1\right )}=0} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class C`]]
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^{2}-x +y \right )^{3}}{x^{2}-x +y +1}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {1}{2 \left (y+x^{2}-x \right )^{2}}-\frac {1}{y+x^{2}-x}\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^{6}-3 x^{5}+3 x^{4} y +3 x^{4}-6 x^{3} y +3 x^{2} y^{2}-x^{3}+3 x^{2} y -3 x \,y^{2}+y^{3}}{2 x^{2}-2 x +2 y +2} \\ \frac {dS}{dR} &= \frac {2}{R} \\ \end{align*}