Link to actual problem [8782] \[ \boxed {\left (x^{2}-1\right ) {y^{\prime }}^{2}-y^{2}=-1} \]
type detected by program
{"first_order_nonlinear_p_but_separable"}
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 [R &= \frac {y^{2}-1}{x^{2}-1}, S \left (R \right ) &= -\frac {\operatorname {arctanh}\left (\frac {\frac {2 \left (y^{2}-1\right ) \left (-1+x \right )}{x^{2}-1}+2}{2 y}\right )}{2}+\frac {\operatorname {arctanh}\left (\frac {-\frac {2 \left (y^{2}-1\right ) \left (1+x \right )}{x^{2}-1}+2}{2 y}\right )}{2}\right ] \\ \end{align*}