\[ y''(x)=\frac {v (v+1) y(x)}{4 x^2}-\frac {(3 x-1) y'(x)}{2 (x-1) x} \] ✓ Mathematica : cpu = 0.0763024 (sec), leaf count = 70
DSolve[Derivative[2][y][x] == (v*(1 + v)*y[x])/(4*x^2) - ((-1 + 3*x)*Derivative[1][y][x])/(2*(-1 + x)*x),y[x],x]
\[\left \{\left \{y(x)\to c_1 i^{-v} x^{-v/2} \, _2F_1\left (\frac {1}{2},-v;\frac {1}{2}-v;x\right )+c_2 i^{v+1} x^{\frac {v+1}{2}} \, _2F_1\left (\frac {1}{2},v+1;v+\frac {3}{2};x\right )\right \}\right \}\] ✓ Maple : cpu = 0.062 (sec), leaf count = 45
dsolve(diff(diff(y(x),x),x) = -1/2/x*(3*x-1)/(x-1)*diff(y(x),x)+1/4*v*(v+1)/x^2*y(x),y(x))
\[y \left (x \right ) = c_{1} x^{-\frac {v}{2}} \hypergeom \left (\left [\frac {1}{2}, -v \right ], \left [\frac {1}{2}-v \right ], x\right )+c_{2} x^{\frac {1}{2}+\frac {v}{2}} \hypergeom \left (\left [\frac {1}{2}, v +1\right ], \left [\frac {3}{2}+v \right ], x\right )\]