Link to actual problem [8925] \[ \boxed {y^{\prime }-\frac {F \left (\frac {y^{2} a +b \,x^{2}}{a}\right ) x}{\sqrt {a}\, y}=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_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 &= -\frac {a}{b x}, \underline {\hspace {1.25 ex}}\eta &= \frac {1}{y}\right ] \\ \left [R &= \frac {a y^{2}+b \,x^{2}}{a}, S \left (R \right ) &= -\frac {b \,x^{2}}{2 a}\right ] \\ \end{align*}