Link to actual problem [9232] \[ \boxed {y^{\prime }-\frac {32 x^{5}+64 x^{6}+64 y^{2} x^{6}+32 y x^{4}+4 x^{2}+64 x^{6} y^{3}+48 y^{2} x^{4}+12 x^{2} y+1}{64 x^{8}}=0} \]
type detected by program
{"abelFirstKind"}
type detected by Maple
[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Abel]
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^{2}, \underline {\hspace {1.25 ex}}\eta &= \frac {1}{2 x}\right ] \\ \left [R &= y+\frac {1}{4 x^{2}}, S \left (R \right ) &= -\frac {1}{x}\right ] \\ \end{align*}