Link to actual problem [3589] \[ \boxed {x \left (1-2 x \right ) y^{\prime }+\left (1+4 x \right ) y-y^{2}=4 x} \]
type detected by program
{"riccati", "exactWithIntegrationFactor", "first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[_rational, [_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 (y -1\right )^{2}}{\left (2 x -1\right ) x}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {\left (2 x -1\right ) x}{y-1}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -\frac {\left (y -1\right ) \left (2 x -y \right )}{2 x -1}\right ] \\ \left [R &= x, S \left (R \right ) &= \left (1-2 x \right ) \left (-\frac {\ln \left (-2 x +y\right )}{2 x -1}+\frac {\ln \left (y-1\right )}{2 x -1}\right )\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 {8 x^{4}-8 x^{3} y +2 x^{2} y^{2}+2 x^{2} y -x \,y^{2}-2 x^{2}+x y}{8 x^{2}-4 x} \\ \frac {dS}{dR} &= -\frac {4}{\left (-1+2 R \right ) R} \\ \end{align*}