\[ y''(x)=-\frac {v (v+1) y(x)}{x^2 \left (x^2-1\right )}-\frac {2 x y'(x)}{x^2-1} \] ✓ Mathematica : cpu = 0.0706361 (sec), leaf count = 86
DSolve[Derivative[2][y][x] == -((v*(1 + v)*y[x])/(x^2*(-1 + x^2))) - (2*x*Derivative[1][y][x])/(-1 + x^2),y[x],x]
\[\left \{\left \{y(x)\to c_1 i^{-v} x^{-v} \, _2F_1\left (\frac {1}{2}-\frac {v}{2},-\frac {v}{2};\frac {1}{2}-v;x^2\right )+c_2 i^{v+1} x^{v+1} \, _2F_1\left (\frac {v}{2}+\frac {1}{2},\frac {v}{2}+1;v+\frac {3}{2};x^2\right )\right \}\right \}\] ✓ Maple : cpu = 0.079 (sec), leaf count = 57
dsolve(diff(diff(y(x),x),x) = -2*x/(x^2-1)*diff(y(x),x)-v*(v+1)/x^2/(x^2-1)*y(x),y(x))
\[y \left (x \right ) = c_{1} \hypergeom \left (\left [-\frac {v}{2}, \frac {1}{2}-\frac {v}{2}\right ], \left [\frac {1}{2}-v \right ], x^{2}\right ) x^{-v}+c_{2} \hypergeom \left (\left [1+\frac {v}{2}, \frac {1}{2}+\frac {v}{2}\right ], \left [\frac {3}{2}+v \right ], x^{2}\right ) x^{v +1}\]