Link to actual problem [9044] \[ \boxed {y^{\prime }-\frac {-\ln \left (x \right )+{\mathrm e}^{\frac {1}{x}}+4 x^{2} y+2 x +2 y^{2} x +2 x^{3}}{\ln \left (x \right )-{\mathrm e}^{\frac {1}{x}}}=0} \]
type detected by program
{"riccati", "first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\).\begin{align*} \\ \\ \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^{2}+2 x y +y^{2}+1 \\ \frac {dS}{dR} &= \frac {2 R}{\ln \left (R \right )-{\mathrm e}^{\frac {1}{R}}} \\ \end{align*}