4.1529   (x2 + 1)y′′(x)+ xy′(x) − 4y(x) = 0

ODE

(x2 + 1)y′′(x)+ xy′(x)− 4y(x) = 0

ODE Classification

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

Book solution method
TO DO

Mathematica
cpu = 0.0131415 (sec), leaf count = 25

{{y(x) → c1cosh(2 sinh−1(x))+ ic2sinh(2sinh−1(x ))}}

Maple
cpu = 0.008 (sec), leaf count = 39

{             (        (         ))         (       (          ))}
                        ----x----                     ----x----
  y(x) =-C1 sin  2arctan  √ − x2-− 1  + C2 cos  2arctan  √ − x2 −-1

Mathematica raw input

DSolve[-4*y[x] + x*y'[x] + (1 + x^2)*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> C[1]*Cosh[2*ArcSinh[x]] + I*C[2]*Sinh[2*ArcSinh[x]]}}

Maple raw input

dsolve((x^2+1)*diff(diff(y(x),x),x)+x*diff(y(x),x)-4*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = _C1*sin(2*arctan(x/(-x^2-1)^(1/2)))+_C2*cos(2*arctan(x/(-x^2-1)^(1/2)))