Link to actual problem [9201] \[ \boxed {y^{\prime }-y^{2}+2 x^{2} y-y^{3}+3 y^{2} x^{2}-3 y x^{4}=-x^{6}+x^{4}+2 x +1} \]
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 ) &= \int _{}^{y}\frac {1}{x^{6}+\left (-3 \textit {\_a} -1\right ) x^{4}+\left (3 \textit {\_a}^{2}+2 \textit {\_a} \right ) x^{2}-\textit {\_a}^{3}-\textit {\_a}^{2}-1}d \textit {\_a}\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 &= 1 \\ \eta &=2 x \\ \frac {dS}{dR} &= \frac {1}{R^{3}+R^{2}+1} \\ \end{align*}