ODE No. 1068

$$v(v+1)y(x)+y^{″}(x)+\cot (x)y^{\prime }(x)=0$$ ✓ Mathematica : cpu = 0.112921 (sec), leaf count = 20

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

$$\{\{y(x)\to c_1{P}_v(\cos (x))+c_2{Q}_v(\cos (x))\}\}$$ ✓ Maple : cpu = 0.196 (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))
 

$$y\left(x\right)=c_1\mathrm{hypergeom}([-\frac{v}{2},\frac{1}{2}+\frac{v}{2}],\left[\frac{1}{2}\right],{\cos }^2\left(x\right))+c_2\cos \left(x\right)\mathrm{hypergeom}([1+\frac{v}{2},\frac{1}{2}-\frac{v}{2}],\left[\frac{3}{2}\right],{\cos }^2\left(x\right))$$