Link to actual problem [9231] \[ \boxed {y^{\prime }-\frac {32 y x^{5}+8 x^{3}+32 x^{5}+64 y^{3} x^{6}+48 y^{2} x^{4}+12 y x^{2}+1}{16 x^{6} \left (4 y x^{2}+1+4 x^{2}\right )}=0} \]
type detected by program
{"unknown"}
type detected by Maple
[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class C`]]
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 {\left (4 x^{2} y +1\right ) \left (16 x^{5} y -16 x^{4} y^{2}+8 x^{5}+4 x^{3}-8 x^{2} y -1\right )}{64 x^{5} \left (4 x^{2} y +4 x^{2}+1\right )}\right ] \\ \\ \end{align*}