Link to actual problem [8978] \[ \boxed {y^{\prime }+\frac {x^{2} \left (a x -2 \sqrt {a \left (a \,x^{4}+8 y\right )}\right )}{2}=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(x)]`]]
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}{x^{2}}, \underline {\hspace {1.25 ex}}\eta &= -\frac {x a}{2}\right ] \\ \left [R &= y+\frac {a \,x^{4}}{8}, S \left (R \right ) &= \frac {x^{3}}{3}\right ] \\ \end{align*}