self-adjoint form
A symmetric matrix is special because it can be flipped inside a dot product without changing anything: v . (A w) = (A v) . w. Self-adjoint form is the trick that makes a differential operator behave the same way. It is the exact shape an operator must wear so that integration by parts moves it cleanly from one factor to the other inside an integral.
Any second-order operator a(x) u'' + b(x) u' + c(x) u can be rewritten, after multiplying by a suitable integrating factor, as -(p u')' + q u — the Sturm-Liouville form. The reason this matters is one line of integration by parts: integral from a to b of v times (-(p u')') dx equals integral of (-(p v')') times u dx plus boundary terms p(v u' - v' u) at the endpoints. If the boundary conditions kill those boundary terms, then the operator satisfies (L u, v) = (u, L v) for the inner product (f, g) = integral of f g dx. The operator is then symmetric, or formally self-adjoint.
This is the structural secret behind the whole Sturm-Liouville theory. Real eigenvalues, orthogonal eigenfunctions, the Rayleigh quotient, the spectral theorem — every good property flows from self-adjointness, exactly as the spectral theorem for symmetric matrices flows from A = A-transpose. Putting an equation into self-adjoint form is therefore the very first move: it reveals whether the linear-algebra machinery applies.
Take x u'' + u' + lambda x u = 0 (Bessel's equation in disguise). Multiplying by nothing extra, group it as -(x u')' = lambda x u — already self-adjoint with p(x) = x, q = 0, weight w(x) = x. The weight x is exactly why Bessel functions are orthogonal with the factor x, i.e. integral of x J_n(...) J_n(...) dx.
Rewriting in self-adjoint form reads off the weight w(x) you must use for orthogonality.
Formally self-adjoint (the operator looks symmetric) is not the same as truly self-adjoint as an operator on a function space — the latter also fixes the precise domain and boundary conditions. For the regular problems in this field the distinction can be ignored, but on singular or unbounded domains it becomes essential.