Link to actual problem [11673] \[ \boxed {y^{\prime }+8 y^{2} x -4 x \left (4 x +1\right ) y=-8 x^{3}-4 x^{2}+1} \]
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*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (2 x -2 y +1\right ) \left (x -y \right )}{2}\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 &=x^{2}-2 x y +y^{2}+\frac {1}{2} x -\frac {1}{2} y \\ \frac {dS}{dR} &= -8 R \\ \end{align*}