Link to actual problem [3983] \[ \boxed {\left (1+\left (y+x \right ) \tan \left (y\right )\right ) y^{\prime }=-1} \]
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 {x +y}{\tan \left (y \right ) x +y \tan \left (y \right )+1}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\ln \left (1+\tan \left (y\right )^{2}\right )}{2}+\ln \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 {x +y}{\tan \left (y \right ) x +y \tan \left (y \right )+1} \\ \frac {dS}{dR} &= 0 \\ \end{align*}