Link to actual problem [3440] \[ \boxed {x y^{\prime }+y-a \,x^{n} \left (x -y\right )^{2}=2 x} \]
type detected by program
{"riccati", "first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _rational, _Riccati]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {1}{x \left (x -y\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 &=-x^{n} a \,x^{2}+2 x^{n} a x y -x^{n} a \,y^{2}+n x -n y -x +y \\ \frac {dS}{dR} &= -\frac {1}{R} \\ \end{align*}