Link to actual problem [9210] \[ \boxed {y^{\prime }-\frac {-2 y x +2 x^{3}-2 x -y^{3}+3 y^{2} x^{2}-3 x^{4} y+x^{6}}{x^{2}-y-1}=0} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, `2nd type`, `class C`]]
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^{2}-y \right )^{3}}{x^{2}-y -1}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {1}{2 \left (-x^{2}+y\right )^{2}}-\frac {1}{-x^{2}+y}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (x^{2}-y \right ) \left (2 x^{5}-4 x^{3} y +2 x \,y^{2}-2 x^{2}+2 y +1\right )}{2 x^{2}-2 y -2}\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^{6}+3 x^{4} y -3 x^{2} y^{2}+y^{3}}{x^{2}-y -1} \\ \frac {dS}{dR} &= -1 \\ \end{align*}