Link to actual problem [3180] \[ \boxed {2 y \left (x +y+2\right )+\left (y^{2}-x^{2}-4 x -1\right ) y^{\prime }=0} \]
type detected by program
{"exactWithIntegrationFactor", "first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _rational]
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}}{x^{2}-y^{2}+4 x +1}\right ] \\ \left [R &= x, S \left (R \right ) &= -y-\frac {x^{2}+4 x +1}{y}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {y \left (x^{2}+2 x y +y^{2}+4 x +1\right )}{x^{2}-y^{2}+4 x +1}\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^{2} y +2 x \,y^{2}+y^{3}+4 x y +4 y^{2}+y}{x^{2}-y^{2}+4 x +1} \\ \frac {dS}{dR} &= 0 \\ \end{align*}