Link to actual problem [4165] \[ \boxed {x \left (-x^{2}+1\right ) {y^{\prime }}^{2}-2 \left (-x^{2}+1\right ) y y^{\prime }+x \left (1-y^{2}\right )=0} \]
type detected by program
{"unknown"}
type detected by Maple
[_rational, [_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 &= \sqrt {1+x}\, x \sqrt {-1+x}, \underline {\hspace {1.25 ex}}\eta &= \sqrt {1+x}\, \sqrt {-1+x}\, y\right ] \\ \left [R &= \frac {y}{x}, S \left (R \right ) &= -\frac {\sqrt {-1+x}\, \sqrt {1+x}\, \arctan \left (\frac {1}{\sqrt {x^{2}-1}}\right )}{\sqrt {x^{2}-1}}\right ] \\ \end{align*}