Link to actual problem [9075] \[ \boxed {y^{\prime }-\frac {\left (y^{2} a +b \,x^{2}\right )^{3} x}{a^{\frac {5}{2}} \left (y^{2} a +b \,x^{2}+a \right ) y}=0} \]
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^{\frac {7}{2}}+3 y^{4} x^{2} a^{\frac {5}{2}} b +3 y^{2} x^{4} b^{2} a^{\frac {3}{2}}+x^{6} b^{3} \sqrt {a}+b \,a^{3} y^{2}+a^{2} b^{2} x^{2}+a^{3} b}{\left (a \,y^{2}+b \,x^{2}+a \right ) y}\right ] \\ \left [R &= x, S \left (R \right ) &= \int _{}^{y}\frac {\left (\textit {\_a}^{2} a +b \,x^{2}+a \right ) \textit {\_a}}{3 \textit {\_a}^{2} x^{4} b^{2} a^{\frac {3}{2}}+3 \textit {\_a}^{4} x^{2} a^{\frac {5}{2}} b +\textit {\_a}^{6} a^{\frac {7}{2}}+b \left (b^{2} x^{6} \sqrt {a}+b \,x^{2} a^{2}+a^{3} \left (\textit {\_a}^{2}+1\right )\right )}d \textit {\_a}\right ] \\ \end{align*}