Link to actual problem [6117] \[ \boxed {\left (y \cos \left (y\right )-\sin \left (y\right )+x \right ) y^{\prime }-y=0} \]
type detected by program
{"exactWithIntegrationFactor", "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 {y^{2}}{-y \cos \left (y \right )+\sin \left (y \right )-x}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {\sin \left (y\right )}{y}+\frac {x}{y}\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 {y^{2}}{-y \cos \left (y \right )+\sin \left (y \right )-x} \\ \frac {dS}{dR} &= 0 \\ \end{align*}