Link to actual problem [9197] \[ \boxed {y^{\prime }-\frac {y \left ({\mathrm e}^{-\frac {x^{2}}{2}} x y+x \,{\mathrm e}^{-\frac {x^{2}}{4}}+2 y^{2} {\mathrm e}^{-\frac {3 x^{2}}{4}}\right ) {\mathrm e}^{\frac {x^{2}}{4}}}{2 y \,{\mathrm e}^{-\frac {x^{2}}{4}}+2}=0} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_Abel, `2nd type`, `class C`], [_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, \underline {\hspace {1.25 ex}}\eta &= \frac {x y}{2}\right ] \\ \left [R &= y \,{\mathrm e}^{-\frac {x^{2}}{4}}, S \left (R \right ) &= x\right ] \\ \end{align*}
My program’s symgen result This shows my program’s found \(\xi ,\eta \) and the corresponding ODE in canonical coordinates \(R,S\).\begin{align*} \xi &= 0 \\ \eta &=\frac {-y^{2} {\mathrm e}^{\frac {x^{2}}{4}} {\mathrm e}^{-\frac {x^{2}}{2}} x -2 \,{\mathrm e}^{-\frac {3 x^{2}}{4}} {\mathrm e}^{\frac {x^{2}}{4}} y^{3}-{\mathrm e}^{-\frac {x^{2}}{4}} {\mathrm e}^{\frac {x^{2}}{4}} x y +x \,y^{2} {\mathrm e}^{-\frac {x^{2}}{4}}+x y}{y \,{\mathrm e}^{-\frac {x^{2}}{4}}+1} \\ \frac {dS}{dR} &= -{\frac {1}{2}} \\ \end{align*}