Link to actual problem [3125] \[ \boxed {\cos \left (y+x \right )-x \sin \left (y+x \right )-x \sin \left (y+x \right ) y^{\prime }=0} \]
type detected by program
{"exact", "first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _exact]
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 {1}{x \sin \left (x +y \right )}\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 {\cos \left (y \right ) \cos \left (x \right )-\sin \left (x \right ) \sin \left (y \right )}{\sin \left (x \right ) \cos \left (y \right )+\cos \left (x \right ) \sin \left (y \right )} \\ \frac {dS}{dR} &= \frac {1}{R} \\ \end{align*}