Link to actual problem [9124] \[ \boxed {y^{\prime }-\frac {2 x \ln \left (\frac {1}{x -1}\right )-\coth \left (\frac {x +1}{x -1}\right )+\coth \left (\frac {x +1}{x -1}\right ) y^{2}-2 \coth \left (\frac {x +1}{x -1}\right ) x^{2} y+\coth \left (\frac {x +1}{x -1}\right ) x^{4}}{\ln \left (\frac {1}{x -1}\right )}=0} \]
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*}