Link to actual problem [9010] \[ \boxed {y^{\prime }-\frac {x +1+2 x^{6} \sqrt {1+4 x^{2} y}}{2 x^{3} \left (x +1\right )}=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\).\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\sqrt {4 x^{2} y +1}}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {4 x^{2} y+1}}{2 x}\right ] \\ \end{align*}