4.1453   − y(x )(4ax2 + n2 − x4)+ x2y′′(x)+ xy′(x) = 0

ODE

− y(x)(4ax2 + n2 − x4)+ x2y′′(x) + xy′(x) = 0

ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica
cpu = 0.131236 (sec), leaf count = 93

((                                                                      ) )
|{|{         n+21 − ix22-( 2)n+12-(  (1-                    2)     n     (  2))|} |}
   y(x) → 2---e-----x------c1U--2(−-2ia-+-n+-1),n+-1,ix--+-c2L-ia− n2− 12-ix-
|(|(                                      x                               |) |)

Maple
cpu = 0.092 (sec), leaf count = 41

{       1 (        (   )          (   ))}
  y(x) = x C2Wia, n2 ix2 + C1Mia, n2 ix2

Mathematica raw input

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

Mathematica raw output

{{y[x] -> (2^((1 + n)/2)*(x^2)^((1 + n)/2)*(C[1]*HypergeometricU[(1 - (2*I)*a + 
n)/2, 1 + n, I*x^2] + C[2]*LaguerreL[-1/2 + I*a - n/2, n, I*x^2]))/(E^((I/2)*x^2
)*x)}}

Maple raw input

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

Maple raw output

y(x) = (_C2*WhittakerW(I*a,1/2*n,I*x^2)+_C1*WhittakerM(I*a,1/2*n,I*x^2))/x