Link to actual problem [9308] \[ \boxed {y^{\prime }-y^{3}-y^{2} x^{2}-\frac {x^{4} y}{3}=\frac {1}{27} x^{6}-\frac {2}{3} x} \]
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 [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (x^{2}+3 y \right )^{3}}{27}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {9}{2 \left (x^{2}+3 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}+12 x^{3} y +18 x \,y^{2}+9\right ) \left (x^{2}+3 y \right )}{54}\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 {3}{2} y^{3}+\frac {3}{2} x^{2} y^{2}+\frac {1}{2} x^{4} y +\frac {1}{18} x^{6} \\ \frac {dS}{dR} &= {\frac {2}{3}} \\ \end{align*}