Link to actual problem [9215] \[ \boxed {y^{\prime }+\frac {\left (-108 x^{\frac {3}{2}}-216-216 y^{2}+72 x^{3} y-6 x^{6}-216 y^{3}+108 x^{3} y^{2}-18 y x^{6}+x^{9}\right ) \sqrt {x}}{216}=0} \]
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 &= \frac {1}{\sqrt {x}}, \underline {\hspace {1.25 ex}}\eta &= \frac {x^{\frac {3}{2}}}{2}\right ] \\ \left [R &= y-\frac {x^{3}}{6}, S \left (R \right ) &= \frac {2 x^{\frac {3}{2}}}{3}\right ] \\ \end{align*}