Fredholm Integral Equations - Sturm-Lioville & Green Function Theory? In an ODE's book one is given a 2nd order ode boundary value problem like
$$y'' + A(x)y' + B(x)y = f(x), y(a) = y_a, y(b) = y_b$$
and might be told to analyze it with a Green function or via Sturm-Liouville theory, and invoke all this orthogonal function theory. For example randomly converting an ode to S-L form. 
It looks to me that GF & S-L theory very naturally arise when you convert the ode to a Fredholm integral equation, as in it looks so obvious these methods are the only thing to use. Further there are some Fredholm theorems that apparently justify what you're doing.
My question, is it 'correct' to view S-L (homogeneous) & GF (inhomogeneous) theory as actually being part of the theory of integral equations, not differential equations, and to view the Fredholm theorems as the real source of justification for things like Fourier, Legendre polynomials being orthogonal and complete, the spectral theorem being an integral equation theorem?
Thanks
 A: If you compare two operation: differentiation and integration, then the clear winner is the integration, you can integrate way more functions than differentiate (moreover, integration is a numerically stable operation, whereas numerical differentiation is an ill posed problem). On the other hand, what is easier to do in Calc, differentiate or integrate? Clearly, find derivatives requires no imagination if you know the rules. A similar thing happens when one talks about S-L problems and the theory behind it. It is easier to find actual eigenfunctions from the ODE, but way better to prove theorems for the integrals. 
Here are some more details.
Sturm-Liouville operator $L$ is symmetric, but unbounded (because it involves differentiation). This means it is quite difficult to prove a lot of things directly for the Sturm-Liouville operator.
On the other hand, we know that if an operator on a separable Hilbert space is compact and self-adjoint, then the whole lot of nice things can be proved, most importantly the completeness of the list of eigenfunctions. 
So, assume that zero is not an eigenvalue of $L$. Then there exists an inverse operator
$$
L^{-1}(f)(y)=\int_a^b f(y)g(x,y)dy,
$$
where $g$ is exactly the Green function, which can be explicitly constructed from the given S-L problem. It is not difficult to prove that, if $g$ is nice enough, the integral above is self-adjoint and compact operator. Hence, using the nice properties of integrals we can infer tons of useful stuff about S-L problems, which appear in applied problems in a natural way.
