Link to actual problem [3386] \[ \boxed {y^{\prime }-{\mathrm e}^{y}=x} \]
type detected by program
{"first order special form ID 1", "first_order_ode_lie_symmetry_lookup"}
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 &= {\mathrm e}^{-\frac {x^{2}}{2}}, \underline {\hspace {1.25 ex}}\eta &= x \,{\mathrm e}^{-\frac {x^{2}}{2}}\right ] \\ \left [R &= -\frac {x^{2}}{2}+y, S \left (R \right ) &= -\frac {i \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, x}{2}\right )}{2}\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 &= {\mathrm e}^{-\frac {x^{2}}{2}} \\ \eta &=x +{\mathrm e}^{-\frac {x^{2}}{2}} \\ \end{align*}