Link to actual problem [9014] \[ \boxed {y^{\prime }-\frac {x^{2}+2 x +1+2 \sqrt {x^{2}+2 x +1-4 y}}{2 \left (x +1\right )}=0} \]
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}}{4}-\frac {x}{2}, S \left (R \right ) &= \frac {\ln \left (1+x \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 &=\frac {-2 x \sqrt {x^{2}+2 x -4 y +1}-2 \sqrt {x^{2}+2 x -4 y +1}}{1+x} \\ \frac {dS}{dR} &= -\frac {1}{2 R +2} \\ \end{align*}