Link to actual problem [9125] \[ \boxed {y^{\prime }-\frac {2 x^{2} \cosh \left (\frac {1}{x -1}\right )-2 x \cosh \left (\frac {1}{x -1}\right )-1+y^{2}-2 x^{2} y+x^{4}-x +y^{2} x -2 x^{3} y+x^{5}}{\left (x -1\right ) \cosh \left (\frac {1}{x -1}\right )}=0} \]
type detected by program
{"riccati"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\).\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {\ln \left (-x^{2}+y-1\right )}{2}-\frac {\ln \left (-x^{2}+y+1\right )}{2}\right ] \\ \end{align*}