If it get's too cumbersome, you can if you wish work on in in Mathematica using this code:
Here's n=3 using the code although I haven't checked it throughly and where I've left the Mathematica-specific notation in since I just copied it directly over so I didn't have to type in the latex.
nval = 1;
myop = D[#1, x] - (a/x)*#1 &
myeqn = FullSimplify[x^(2*nval)*
Nest[myop, f[x], nval]] == k*f[x]
FullSimplify[DSolve[myeqn, f, x]]
As a check of the code, here is n=2 which I believe is the same as Danny's solution:
Double-check everything though ok.