\[ \sqrt {x^2-1} y'(x)-\sqrt {y(x)^2-1}=0 \] ✓ Mathematica : cpu = 0.0782209 (sec), leaf count = 173
DSolve[-Sqrt[-1 + y[x]^2] + Sqrt[-1 + x^2]*Derivative[1][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to -\frac {1}{2} e^{-c_1} \sqrt {2 x^2+2 e^{4 c_1} x^2-2 \sqrt {(x-1) (x+1)} x+2 e^{4 c_1} \sqrt {(x-1) (x+1)} x-1+2 e^{2 c_1}-e^{4 c_1}}\right \},\left \{y(x)\to \frac {1}{2} e^{-c_1} \sqrt {2 x^2+2 e^{4 c_1} x^2-2 \sqrt {(x-1) (x+1)} x+2 e^{4 c_1} \sqrt {(x-1) (x+1)} x-1+2 e^{2 c_1}-e^{4 c_1}}\right \}\right \}\] ✓ Maple : cpu = 0.007 (sec), leaf count = 29
dsolve((x^2-1)^(1/2)*diff(y(x),x)-(y(x)^2-1)^(1/2) = 0,y(x))
\[\ln \left (x +\sqrt {x^{2}-1}\right )-\ln \left (y \left (x \right )+\sqrt {y \left (x \right )^{2}-1}\right )+c_{1} = 0\]