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**"Self-Adjointness" on Differential Equations**

Hey, Im just wondering why is that we always try to look for self-adjoints Differential Equations. I mean I know the advantages of having self-adjoint operators, i.e, they have real eigenvalues, eigenfunctions are orthogonal and form a complete set. But, Im having a hard time trying to relate this to solving actual differential equations, can you give me quick tips or ideas on this? or what should I look for?... Thanks.