Link to actual problem [8940] \[ \boxed {y^{\prime }+\left (-{\mathrm e}^{-x^{2}}+x^{2} {\mathrm e}^{-x^{2}}-F \left (y-\frac {x^{2} {\mathrm e}^{-x^{2}}}{2}\right )\right ) x=0} \]
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 &= -\left (x^{2}-1\right ) {\mathrm e}^{-x^{2}}\right ] \\ \left [R &= y-\frac {x^{2} {\mathrm e}^{-x^{2}}}{2}, S \left (R \right ) &= \frac {x^{2}}{2}\right ] \\ \end{align*}