\[ \left (x^2-1\right ) y''(x)-v (v+1) y(x)=0 \] ✓ Mathematica : cpu = 0.039433 (sec), leaf count = 58
DSolve[-(v*(1 + v)*y[x]) + (-1 + x^2)*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_1 \operatorname {Hypergeometric2F1}\left (-\frac {v}{2}-\frac {1}{2},\frac {v}{2},\frac {1}{2},x^2\right )+i c_2 x \operatorname {Hypergeometric2F1}\left (\frac {v}{2}+\frac {1}{2},-\frac {v}{2},\frac {3}{2},x^2\right )\right \}\right \}\] ✓ Maple : cpu = 0.083 (sec), leaf count = 52
dsolve((x^2-1)*diff(diff(y(x),x),x)-v*(v+1)*y(x)=0,y(x))
\[y \left (x \right ) = -\left (x -1\right ) \left (1+x \right ) \left (\operatorname {hypergeom}\left (\left [1-\frac {v}{2}, \frac {3}{2}+\frac {v}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right ) c_{2} x +c_{1} \operatorname {hypergeom}\left (\left [1+\frac {v}{2}, \frac {1}{2}-\frac {v}{2}\right ], \left [\frac {1}{2}\right ], x^{2}\right )\right )\]