Link to actual problem [9142] \[ \boxed {y^{\prime }-\frac {\left (1+2 y\right ) \left (y+1\right )}{x \left (-2 y-2+x +2 x y\right )}=0} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class B`]]
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 &= -x \left (y +1\right ), \underline {\hspace {1.25 ex}}\eta &= y^{2}+\frac {3}{2} y +\frac {1}{2}\right ] \\ \left [R &= \frac {\left (2 y+1\right ) x}{2}, S \left (R \right ) &= -2 \ln \left (y+1\right )+2 \ln \left (2 y+1\right )\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 +2 x^{2}-4 x y -4 x} \\ \frac {dS}{dR} &= 0 \\ \end{align*}