The problem is $$x^{2n} \left (\frac{d}{dx} - \frac{1}{a}\right )^{n} y = ky$$
I'm pretty sure the first step is to take $x = e^{t}$ and play with the operators like so $$\frac{d}{dt} = \frac{dx}{dt} \frac{d}{dx} = e^{t}\frac{d}{dx}$$
I'm not used to using operators in this fashion to solve ODEs so here is where i got lost but i think you can convert the operator part of the problem to
$$\left (\frac{e^{t}}{e^{t}}\frac{d}{dt}- \frac{a}{e^{t}}\right )^{n}$$ Can i just pull $e^{-nt}$ out of the operator? How does this work? any help or suggestions would be greatly appreciated.