Link to actual problem [9307] \[ \boxed {y^{\prime }-y^{3}+3 y^{2} x^{2}-3 x^{4} y=-x^{6}+2 x} \]
type detected by program
{"abelFirstKind", "first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _Abel]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {1}{2 \left (x^{2}-y\right )^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (2 x^{5}-4 x^{3} y +2 x \,y^{2}+1\right ) \left (x^{2}-y \right )}{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 &=x^{6}-3 x^{4} y +3 x^{2} y^{2}-y^{3} \\ \frac {dS}{dR} &= -1 \\ \end{align*}