Link to actual problem [9261] \[ \boxed {y^{\prime }-\frac {\left ({\mathrm e}^{-\frac {y}{x}} y x +{\mathrm e}^{-\frac {y}{x}} y+{\mathrm e}^{-\frac {y}{x}} x^{2}+{\mathrm e}^{-\frac {y}{x}} x +x \right ) {\mathrm e}^{\frac {y}{x}}}{x \left (x +1\right )}=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(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 &= 1+x, \underline {\hspace {1.25 ex}}\eta &= x +y +1+\frac {y}{x}\right ] \\ \left [R &= -\frac {-y+x \ln \left (x \right )}{x}, S \left (R \right ) &= \ln \left (1+x \right )\right ] \\ \end{align*}