Link to actual problem [9228] \[ \boxed {y^{\prime }-\frac {\left (-256 y a \,x^{2}-32 a^{2} x^{6}-256 a \,x^{2}+512 y^{3}+192 y^{2} a \,x^{4}+24 y a^{2} x^{8}+a^{3} x^{12}\right ) x}{512 y+64 a \,x^{4}+512}=0} \]
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 (a \,x^{4}+8 y \right )^{3}}{32768 a \,x^{4}+262144 y +262144}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {4096}{a \,x^{4}+8 y}-\frac {16384}{\left (a \,x^{4}+8 y\right )^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (a \,x^{4}+8 y \right ) \left (a^{2} x^{10}+16 a \,x^{6} y +16 a \,x^{4}+64 x^{2} y^{2}+128 y +64\right )}{32768 a \,x^{4}+262144 y +262144}\right ] \\ \\ \end{align*}