\[ y''(x)=-\frac {y(x) \left (-4 n^2 x-v (v+1) (x-1)^2\right )}{4 (x-1)^2 x^2}-\frac {(3 x-1) y'(x)}{2 (x-1) x} \] ✓ Mathematica : cpu = 0.274974 (sec), leaf count = 217
DSolve[Derivative[2][y][x] == -1/4*((-(v*(1 + v)*(-1 + x)^2) - 4*n^2*x)*y[x])/((-1 + x)^2*x^2) - ((-1 + 3*x)*Derivative[1][y][x])/(2*(-1 + x)*x),y[x],x]
\[\left \{\left \{y(x)\to c_2 (-1)^{\frac {1}{2} (-2 v-3)+1} x^{\frac {1}{4} (-2 v-3)+1} e^{\frac {1}{4} (-2 \log (1-x)-\log (x))} (x-1)^{\frac {1}{2} \left (n+\frac {1}{2} (2 n+1)+\frac {1}{2} (-2 v-3)+v+2\right )} \, _2F_1\left (\frac {1}{2} (2 n+1)+\frac {1}{2} (-2 v-3)+1,n+\frac {1}{2} (-2 v-3)+v+2;\frac {1}{2} (-2 v-3)+2;x\right )+c_1 x^{\frac {1}{4} (2 v+3)} e^{\frac {1}{4} (-2 \log (1-x)-\log (x))} (x-1)^{\frac {1}{2} \left (n+\frac {1}{2} (2 n+1)+\frac {1}{2} (-2 v-3)+v+2\right )} \, _2F_1\left (\frac {1}{2} (2 n+1),n+v+1;\frac {1}{2} (2 v+3);x\right )\right \}\right \}\] ✓ Maple : cpu = 0.064 (sec), leaf count = 68
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-1)^2-4*n^2*x)/x^2/(x-1)^2*y(x),y(x))
\[y \left (x \right ) = \left (x -1\right )^{-n} \left (x^{-\frac {v}{2}} \hypergeom \left (\left [-v -n , -n +\frac {1}{2}\right ], \left [\frac {1}{2}-v \right ], x\right ) c_{1}+x^{\frac {1}{2}+\frac {v}{2}} \hypergeom \left (\left [v -n +1, -n +\frac {1}{2}\right ], \left [\frac {3}{2}+v \right ], x\right ) c_{2}\right )\]