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The original question actually goes like this:

verify that y(x) = sin (x

^{2}) is in the kernel of L,

L = D

^{2}- x

^{-1}D + 4x

^{2}, where D is a differetiation operator.

so what I have so far is this:

Ly = 0

when I distribute I get this DE and get stuck with it:

y'' - x

^{-1}y' + 4x

^{2}y = 0

Thanks for any help.