2.1068   ODE No. 1068

\[ v (v+1) y(x)+y''(x)+\cot (x) y'(x)=0 \] Mathematica : cpu = 0.079406 (sec), leaf count = 20

DSolve[v*(1 + v)*y[x] + Cot[x]*Derivative[1][y][x] + Derivative[2][y][x] == 0,y[x],x]
 

\[\{\{y(x)\to c_1 \operatorname {LegendreP}(v,\cos (x))+c_2 \operatorname {LegendreQ}(v,\cos (x))\}\}\] Maple : cpu = 0.309 (sec), leaf count = 45

dsolve(diff(diff(y(x),x),x)+diff(y(x),x)*cot(x)+v*(v+1)*y(x)=0,y(x))
 

\[y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [-\frac {v}{2}, \frac {1}{2}+\frac {v}{2}\right ], \left [\frac {1}{2}\right ], \cos \left (x \right )^{2}\right )+c_{2} \cos \left (x \right ) \operatorname {hypergeom}\left (\left [1+\frac {v}{2}, \frac {1}{2}-\frac {v}{2}\right ], \left [\frac {3}{2}\right ], \cos \left (x \right )^{2}\right )\]