Link to actual problem [9216] \[ \boxed {y^{\prime }-\frac {\left (a^{3}+y^{4} a^{3}+2 a^{2} b \,x^{2} y^{2}+a \,b^{2} x^{4}+y^{6} a^{3}+3 y^{4} a^{2} b \,x^{2}+3 y^{2} a \,b^{2} x^{4}+b^{3} x^{6}\right ) x}{a^{\frac {7}{2}} y}=0} \]
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 {y^{6} a^{3}+3 a^{2} b \,x^{2} y^{4}+3 a \,b^{2} x^{4} y^{2}+b^{3} x^{6}+a^{3} y^{4}+2 a^{2} y^{2} b \,x^{2}+a \,x^{4} b^{2}+a^{3}+a^{\frac {5}{2}} b}{y}\right ] \\ \left [R &= x, S \left (R \right ) &= \int _{}^{y}\frac {\textit {\_a}}{a^{\frac {5}{2}} b +\left (\textit {\_a}^{6}+\textit {\_a}^{4}+1\right ) a^{3}+3 x^{2} \left (\textit {\_a}^{2}+\frac {2}{3}\right ) \textit {\_a}^{2} b \,a^{2}+3 x^{4} b^{2} \left (\textit {\_a}^{2}+\frac {1}{3}\right ) a +b^{3} x^{6}}d \textit {\_a}\right ] \\ \end{align*}