Link to actual problem [9152] \[ \boxed {y^{\prime }-\frac {y}{x \left (-1+x y+x y^{3}+y^{4} x \right )}=0} \]
type detected by program
{"exactWithIntegrationFactor", "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 {y^{2} x}{x \,y^{4}+x \,y^{3}+x y -1} \\ \frac {dS}{dR} &= 0 \\ \end{align*}