Link to actual problem [9162] \[ \boxed {y^{\prime }-\frac {\left (1+2 y\right ) \left (y+1\right )}{x \left (-2 y-2+x y^{3}+2 y^{4} x \right )}=0} \]
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 [\underline {\hspace {1.25 ex}}\xi &= \frac {x^{2} \left (2 y +1\right )}{2}, \underline {\hspace {1.25 ex}}\eta &= 0\right ] \\ \left [R &= y, S \left (R \right ) &= -\frac {2}{\left (2 y+1\right ) 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 {-4 x^{2} y^{3}-8 x^{2} y^{2}-5 x^{2} y -x^{2}}{4 x^{2} y^{4}+2 x^{2} y^{3}-4 x y -4 x} \\ \frac {dS}{dR} &= 0 \\ \end{align*}