Link to actual problem [8967] \[ \boxed {y^{\prime }-\frac {{\mathrm e}^{\frac {2 x}{3}}}{y \,{\mathrm e}^{-\frac {2 x}{3}}+1}=0} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], [_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 &= 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 &= 0 \\ \eta &=\frac {2 \,{\mathrm e}^{-\frac {2 x}{3}} y^{2}-3 \,{\mathrm e}^{\frac {2 x}{3}}+2 y}{2 y \,{\mathrm e}^{-\frac {2 x}{3}}+2} \\ \frac {dS}{dR} &= -{\frac {1}{3}} \\ \end{align*}