Link to actual problem [8992] \[ \boxed {y^{\prime }+\frac {x^{2}-1-4 \sqrt {x^{2}-2 x +1+8 y}}{4 x +4}=0} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(x)]`]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\).\begin{align*} \\ \left [R &= y+\frac {x^{2}}{8}-\frac {x}{4}, S \left (R \right ) &= -\frac {\ln \left (-x -1\right )}{4}\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 {4 x \sqrt {x^{2}-2 x +8 y +1}+4 \sqrt {x^{2}-2 x +8 y +1}}{1+x} \\ \frac {dS}{dR} &= \frac {1}{4 R +4} \\ \end{align*}