Link to actual problem [9163] \[ \boxed {y^{\prime }-\frac {1+2 \sqrt {1+4 x^{2} y}\, x^{3}+2 x^{5} \sqrt {1+4 x^{2} y}+2 x^{6} \sqrt {1+4 x^{2} y}}{2 x^{3}}=0} \]
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 {2 \sqrt {4 x^{2} y +1}}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {4 x^{2} y+1}}{4 x}\right ] \\ \end{align*}