4.1315   y′′(x)+ 2 cot(x)y′(x)+ 3y(x) = 0

ODE

y′′(x) + 2cot(x)y′(x)+ 3y(x) = 0

ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica
cpu = 0.112467 (sec), leaf count = 42

{{                                } }
          ---c2e3ix----    − 2ix
   y(x) → 2(− 1+ e2ix) + c1e csc(x)

Maple
cpu = 0.051 (sec), leaf count = 22

{                             }
 y(x) = -C1sin(2x)+-C2-cos(2x)
                sin (x)

Mathematica raw input

DSolve[3*y[x] + 2*Cot[x]*y'[x] + y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (E^((3*I)*x)*C[2])/(2*(-1 + E^((2*I)*x))) + (C[1]*Csc[x])/E^((2*I)*x)}
}

Maple raw input

dsolve(diff(diff(y(x),x),x)+2*cot(x)*diff(y(x),x)+3*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = (_C1*sin(2*x)+_C2*cos(2*x))/sin(x)