4.1216   y(x)(a0 +a1x + a2x2 + a3x3 + a4x4 + x8) +y′′(x) = 0

ODE

y(x)(a0 + a1x + a2x2 + a3x3 + a4x4 + x8) +y′′(x) = 0

ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica
cpu = 2.13896 (sec), leaf count = 0 , DifferentialRoot result

{{y(x) → DifferentialRoot({y,x},{(x8 + a4x4 + a3x3 +a2x2 + a1x + a0)y(x)+ y′′(x) = 0,y(0) = c1,y′(0) = c2} )(x)}}

Maple
cpu = 1.177 (sec), leaf count = 0 , result contains DESol

{             ({   2        ∑m             }        ) }
 y (x) = DESol    -d2 Y (x)+    a(n)xn Y (x) ,{-Y (x )}
                 dx         n=0

Mathematica raw input

DSolve[(a0 + a1*x + a2*x^2 + a3*x^3 + a4*x^4 + x^8)*y[x] + y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> DifferentialRoot[Function[{\[FormalY], \[FormalX]}, {(\[FormalX]^8 + a
0 + \[FormalX]*a1 + \[FormalX]^2*a2 + \[FormalX]^3*a3 + \[FormalX]^4*a4)*\[Forma
lY][\[FormalX]] + Derivative[2][\[FormalY]][\[FormalX]] == 0, \[FormalY][0] == C
[1], Derivative[1][\[FormalY]][0] == C[2]}]][x]}}

Maple raw input

dsolve(diff(diff(y(x),x),x)+sum(a(n)*x^n,n = 0 .. m)*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = DESol({diff(diff(_Y(x),x),x)+sum(a(n)*x^n,n = 0 .. m)*_Y(x)},{_Y(x)})