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When is the following operator self-adjoint?(Is there a difference between self-adjoint and Hermitian?)

$O:= \sum_{n=0}^4 f_n(x){d^n\over dx^n}$ subjected to boundary conditions $y(0)=y'(0)=y(1)=y'(1)=0$ and where $f_n$'s are real functions.

Thanks.

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    On which space is $y$?2011-09-27
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    @DavideGiraudo: I beleive it is the space of functions that satisfy those boundary conditions?2011-09-27
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    If I remember correctly, in the case of unbounded operators (as in here), self-adjoint is not the same as hermitian/symmetric.2011-09-27

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