Self-Adjointness on Differential Equations

[mefistofeles]

"Self-Adjointness" on Differential Equations

Hey, I am 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, I am 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.

 

[HallsofIvy]

Those are the reasons! The differential operators of a self adjoint differential equations are self adjoint transformations so there exist a basis for the solution space consisting of "eigenfunctions" of those operators and it becomes easy to find the solution.