Link to actual problem [3212] \[ \boxed {y+\left (x -y \left (y+1\right )^{2}\right ) y^{\prime }=-1} \]
type detected by program
{"exact", "differentialType", "first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[_exact, _rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]
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 {1}{-y^{3}-2 y^{2}+x -y}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {y^{4}}{4}-\frac {2 y^{3}}{3}+x y-\frac {y^{2}}{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 {-3 y^{4}-8 y^{3}+12 x y -6 y^{2}+12 x +1}{-3 y^{3}-6 y^{2}+3 x -3 y} \\ \frac {dS}{dR} &= 0 \\ \end{align*}