Link to actual problem [3311] \[ \boxed {y^{\prime }-\left (2 x^{2}-y\right ) y=1+x \left (-x^{3}+2\right )} \]
type detected by program
{"riccati", "first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _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 {\ln \left (-x^{2}+y-1\right )}{2}-\frac {\ln \left (-x^{2}+y+1\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^{4}-2 x^{2} y +y^{2}-1 \\ \frac {dS}{dR} &= -1 \\ \end{align*}