Link to actual problem [9207] \[ \boxed {y^{\prime }-\frac {\left (-256 a \,x^{2}+512+512 y^{2}+128 y a \,x^{4}+8 a^{2} x^{8}+512 y^{3}+192 y^{2} a \,x^{4}+24 y a^{2} x^{8}+a^{3} x^{12}\right ) x}{512}=0} \]
type detected by program
{"abelFirstKind"}
type detected by Maple
[[_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 &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {1}{512}+\frac {\left (a \,x^{4}+8 y +8\right ) \left (a \,x^{4}+8 y \right )^{2}}{262144}\right ] \\ \left [R &= x, S \left (R \right ) &= \int _{}^{y}\frac {1}{\frac {a^{3} x^{12}}{262144}+\frac {3 \left (\textit {\_a} +\frac {1}{3}\right ) x^{8} a^{2}}{32768}+\frac {3 \left (\textit {\_a} +\frac {2}{3}\right ) \textit {\_a} \,x^{4} a}{4096}+\frac {\textit {\_a}^{3}}{512}+\frac {\textit {\_a}^{2}}{512}+\frac {1}{512}}d \textit {\_a}\right ] \\ \end{align*}