Link to actual problem [8498] \[ \boxed {\left (x -a \right ) \left (x -b \right ) y^{\prime }+y^{2}+k \left (y+x -a \right ) \left (y+x -b \right )=0} \]
type detected by program
{"riccati", "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 \(R,S\).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 {a b \,k^{2}-a \,k^{2} x -a \,k^{2} y -b \,k^{2} x -b \,k^{2} y +k^{2} x^{2}+2 k^{2} x y +k^{2} y^{2}-a k y -b k y +2 k x y +2 k \,y^{2}+y^{2}}{k^{2}} \\ \frac {dS}{dR} &= -\frac {k^{2}}{\left (-R +a \right ) \left (-R +b \right ) \left (k +1\right )} \\ \end{align*}