Link to actual problem [9031] \[ \boxed {y^{\prime }-\left (1+y^{2} {\mathrm e}^{-\frac {4 x}{3}}+{\mathrm e}^{-2 x} y^{3}\right ) {\mathrm e}^{\frac {2 x}{3}}=0} \]
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 &= 1, \underline {\hspace {1.25 ex}}\eta &= \frac {2 y}{3}\right ] \\ \left [R &= y \,{\mathrm e}^{-\frac {2 x}{3}}, S \left (R \right ) &= 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 &= {\frac {3}{2}} \\ \eta &=y \\ \frac {dS}{dR} &= \frac {2}{3 R^{3}+3 R^{2}-2 R +3} \\ \end{align*}