\[ y'(x)-\frac {\sqrt {y(x)^2-1}}{\sqrt {x^2-1}}=0 \] ✓ Mathematica : cpu = 0.0583219 (sec), leaf count = 173
\[\left \{\left \{y(x)\to -\frac {1}{2} e^{-c_1} \sqrt {2 e^{4 c_1} x^2+2 e^{4 c_1} \sqrt {(x-1) (x+1)} x+2 e^{2 c_1}-e^{4 c_1}+2 x^2-2 \sqrt {(x-1) (x+1)} x-1}\right \},\left \{y(x)\to \frac {1}{2} e^{-c_1} \sqrt {2 e^{4 c_1} x^2+2 e^{4 c_1} \sqrt {(x-1) (x+1)} x+2 e^{2 c_1}-e^{4 c_1}+2 x^2-2 \sqrt {(x-1) (x+1)} x-1}\right \}\right \}\] ✓ Maple : cpu = 0.023 (sec), leaf count = 29
\[ \left \{ \ln \left ( x+\sqrt {{x}^{2}-1} \right ) -\ln \left ( y \left ( x \right ) +\sqrt { \left ( y \left ( x \right ) \right ) ^{2}-1} \right ) +{\it \_C1}=0 \right \} \]
\begin {equation} y^{\prime }=\pm \frac {\sqrt {y^{2}-1}}{\sqrt {x^{2}-1}}\tag {1} \end {equation}
Separable. For the positive case
\begin {align*} \frac {dy}{dx}\frac {1}{\sqrt {y^{2}-1}} & =\frac {1}{\sqrt {x^{2}-1}}\\ \frac {dy}{\left ( y^{2}-1\right ) ^{\frac {1}{2}}} & =\frac {dx}{\left ( x^{2}-1\right ) ^{\frac {1}{2}}} \end {align*}
Integrating
\[ \int \frac {dy}{\left ( y^{2}-1\right ) ^{\frac {1}{2}}}=\int \frac {dx}{\left ( x^{2}-1\right ) ^{\frac {1}{2}}}+C \]
But \(\int \frac {dy}{\left ( y^{2}-1\right ) ^{\frac {1}{2}}}=\tanh ^{-1}\frac {y}{\left ( y^{2}-1\right ) ^{\frac {1}{2}}}=\ln \left ( y+\sqrt {y^{2}-1}\right ) \), hence
\[ \ln \left ( y+\sqrt {y^{2}-1}\right ) =\ln \left ( x+\sqrt {x^{2}-1}\right ) +C \]
For the negative case
\begin {align*} \frac {dy}{dx}\frac {1}{\sqrt {y^{2}-1}} & =-\frac {1}{\sqrt {x^{2}-1}}\\ \frac {dy}{\left ( y^{2}-1\right ) ^{\frac {1}{2}}} & =-\frac {dx}{\left ( x^{2}-1\right ) ^{\frac {1}{2}}} \end {align*}
Integrating
\[ \int \frac {dy}{\left ( y^{2}-1\right ) ^{\frac {1}{2}}}=-\int \frac {dx}{\left ( x^{2}-1\right ) ^{\frac {1}{2}}}+C \]
But \(\int \frac {dy}{\left ( y^{2}-1\right ) ^{\frac {1}{2}}}=\tanh ^{-1}\frac {y}{\left ( y^{2}-1\right ) ^{\frac {1}{2}}}=\ln \left ( y+\sqrt {y^{2}-1}\right ) \), hence
\[ \ln \left ( y+\sqrt {y^{2}-1}\right ) =-\ln \left ( x+\sqrt {x^{2}-1}\right ) +C \]
Therefore
\[ \ln \left ( y+\sqrt {y^{2}-1}\right ) =\pm \ln \left ( x+\sqrt {x^{2}-1}\right ) +C \]