Link to actual problem [7384] \[ \boxed {y^{\prime }-10 \,{\mathrm e}^{x +y}=x^{2}} \]
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 &= \frac {{\mathrm e}^{-\frac {1}{3} x^{3}-x}}{10}, \underline {\hspace {1.25 ex}}\eta &= \frac {x^{2} {\mathrm e}^{-\frac {1}{3} x^{3}-x}}{10}\right ] \\ \left [R &= -\frac {x^{3}}{3}+y, S \left (R \right ) &= \int 10 \,{\mathrm e}^{x +\frac {1}{3} x^{3}}d 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 &= \frac {{\mathrm e}^{-\frac {1}{3} x^{3}-x}}{10} \\ \eta &=x^{2}+\frac {{\mathrm e}^{-\frac {1}{3} x^{3}-x}}{10} \\ \end{align*}