Link to actual problem [9264] \[ \boxed {y^{\prime }-\frac {-3 x^{2} y-2 x^{3}-2 x -y^{2} x -y+y^{3} x^{3}+3 y^{2} x^{4}+3 x^{5} y+x^{6}}{x \left (x y+x^{2}+1\right )}=0} \]
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 {x^{2} \left (x +y \right )^{3}}{x^{2}+x y +1}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {-\frac {x}{x +y}-\frac {1}{2 \left (x +y\right )^{2}}}{x^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (x +y \right ) \left (2 x^{5}+4 x^{4} y +2 x^{3} y^{2}+2 x^{2}+2 x y +1\right )}{2 x^{2}+2 x y +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 {2 x^{6}+6 x^{5} y +6 x^{4} y^{2}+2 x^{3} y^{3}+x^{5}+3 x^{4} y +3 x^{3} y^{2}+x^{2} y^{3}+2 x^{3}+4 x^{2} y +2 x \,y^{2}+x +y}{x^{2}+x y +1} \\ \frac {dS}{dR} &= -\frac {1}{R} \\ \end{align*}