Link to actual problem [6201] \[ \boxed {\frac {-y^{\prime } x +y}{\left (x +y\right )^{2}}+y^{\prime }=1} \]
type detected by program
{"exact", "first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _exact, _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 {\left (x +y \right )^{2}}{x^{2}+2 x y +y^{2}-x}\right ] \\ \left [R &= x, S \left (R \right ) &= y+\frac {x}{x +y}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -\frac {\left (x +y \right ) \left (x^{2}-y^{2}-x \right )}{x^{2}+2 x y +y^{2}-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 &= 0 \\ \eta &=\frac {-2 x^{3}-2 x^{2} y +2 x \,y^{2}+2 y^{3}+x^{2}-y^{2}}{x^{2}+2 x y +y^{2}-x} \\ \frac {dS}{dR} &= 0 \\ \end{align*}