Link to actual problem [8917] \[ \boxed {y^{\prime }+\frac {\left (a \,x^{2}-2 F \left (y+\frac {a \,x^{4}}{8}\right )\right ) x}{2}=0} \]
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}, \underline {\hspace {1.25 ex}}\eta &= -\frac {x^{2} a}{2}\right ] \\ \left [R &= y+\frac {a \,x^{4}}{8}, S \left (R \right ) &= \frac {x^{2}}{2}\right ] \\ \end{align*}