4.1215   axky(x)+ y′′(x) = 0

ODE

axky(x)+ y′′(x) = 0

ODE Classification

[[_Emden, _Fowler]]

Book solution method
TO DO

Mathematica
cpu = 0.0699298 (sec), leaf count = 112

{{                         (                 (         )                     (         ) )} }
               − 1-√ ---1--     (k + 1)        2√ax k2+1       (     1  )       2√ax-k2+1
   y(x) → (k + 2) k+2  xa2k+4 c1Γ  k-+-2 J −k1+2  -k-+-2--  + c2Γ  1 + k+-2- Jk1+2  --k+-2--

Maple
cpu = 0.141 (sec), leaf count = 59

{         -(        ( √ --k∕2+1 )             ( √ --k∕2+1 )   )}
  y(x) = √ x Y (k+2)−1 2--ax----- -C2 +J(k+2)−1  2--ax----- -C1
                        k +2                     k +2

Mathematica raw input

DSolve[a*x^k*y[x] + y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (a^(4 + 2*k)^(-1)*Sqrt[x]*(BesselJ[-(2 + k)^(-1), (2*Sqrt[a]*x^(1 + k/
2))/(2 + k)]*C[1]*Gamma[(1 + k)/(2 + k)] + BesselJ[(2 + k)^(-1), (2*Sqrt[a]*x^(1
 + k/2))/(2 + k)]*C[2]*Gamma[1 + (2 + k)^(-1)]))/(2 + k)^(2 + k)^(-1)}}

Maple raw input

dsolve(diff(diff(y(x),x),x)+a*x^k*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = x^(1/2)*(BesselY(1/(k+2),2*a^(1/2)*x^(1/2*k+1)/(k+2))*_C2+BesselJ(1/(k+2)
,2*a^(1/2)*x^(1/2*k+1)/(k+2))*_C1)