ODE No. 1416

$$y^{″}(x)=(n-v)(n+v+1)y(x)-(2n+1)\cot (x)y^{\prime }(x)$$ ✓ Mathematica : cpu = 0.163673 (sec), leaf count = 46

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

$$\left\{\{y(x)\to c_1{({\cos }^2(x)-1)}^{-n/2}{P}_v^n(\cos (x))+c_2{({\cos }^2(x)-1)}^{-n/2}{Q}_v^n(\cos (x))\}\right\}$$ ✓ Maple : cpu = 0.167 (sec), leaf count = 26

dsolve(diff(diff(y(x),x),x) = -(2*n+1)*cos(x)/sin(x)*diff(y(x),x)-(v+n+1)*(v-n)*y(x),y(x))
 

$$y\left(x\right)=({\sin }^{-n}\left(x\right))(\mathrm{LegendreQ}(v,n,\cos \left(x\right))c_2+\mathrm{LegendreP}(v,n,\cos \left(x\right))c_1)$$