Link to actual problem [8981] \[ \boxed {y^{\prime }-\frac {\left (y^{2} a +b \,x^{2}\right )^{2} x}{a^{\frac {5}{2}} y}=0} \]
type detected by program
{"unknown"}
type detected by Maple
[_rational, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]
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 {a^{2} y^{4}+2 a b \,x^{2} y^{2}+b^{2} x^{4}+b \,a^{\frac {3}{2}}}{y}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\arctan \left (\frac {2 y^{2} a^{2}+2 a b \,x^{2}}{2 \sqrt {a^{\frac {7}{2}} b}}\right )}{2 \sqrt {a^{\frac {7}{2}} b}}\right ] \\ \end{align*}