Link to actual problem [7385] \[ \boxed {y^{\prime }-{\mathrm e}^{x +y} x=\sin \left (x \right )} \]
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}^{-x +\cos \left (x \right )}}{x}, \underline {\hspace {1.25 ex}}\eta &= \frac {\sin \left (x \right ) {\mathrm e}^{-x +\cos \left (x \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 &= \frac {{\mathrm e}^{-x +\cos \left (x \right )}}{x} \\ \eta &=\sin \left (x \right )+\frac {{\mathrm e}^{-x +\cos \left (x \right )}}{x} \\ \end{align*}