Link to actual problem [9111] \[ \boxed {y^{\prime }-\frac {y \left (y+1\right )}{x \left (-y-1+y^{4} x \right )}=0} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[_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 [R &= y, S \left (R \right ) &= -\frac {1}{x y}\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 \,y^{3}-x \,y^{2}}{x \,y^{4}-y -1} \\ \frac {dS}{dR} &= 0 \\ \end{align*}