Link to actual problem [108] \[ \boxed {\left ({\mathrm e}^{y}+x \right ) y^{\prime }-x \,{\mathrm e}^{-y}=-1} \]
type detected by program
{"exact", "first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries]]
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 &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{-y}}{{\mathrm e}^{y}+x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{2 y}}{2}+{\mathrm e}^{y} 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 {{\mathrm e}^{2 y}+2 \,{\mathrm e}^{y} x -x^{2}}{{\mathrm e}^{2 y}+{\mathrm e}^{y} x} \\ \frac {dS}{dR} &= 0 \\ \end{align*}