What is the Role of Weight Functions in Solving Differential Equations?

[id the sloth]

I've read a few books on solving differential equations and they all talk about inner products with respect to weight functions. The examples always read something like that "Using the weight function w(x) = blah blah show that the solutions of the differential equations are orthogonal"

We're always given the weight function. I was wondering if there was a general method to finding the weight function?

 

[Akater]

Yes, there are ways to determine weight. For example, Vilenkin does that a couple of times in his Special functions and the theory of group representations.

 

[LCKurtz]

I've read a few books on solving differential equations and they all talk about inner products with respect to weight functions. The examples always read something like that "Using the weight function w(x) = blah blah show that the solutions of the differential equations are orthogonal"

We're always given the weight function. I was wondering if there was a general method to finding the weight function?

Many of the common orthogonal families of functions are solutions to Sturm-Liouville boundary value problems. These are eigenvalue problems with associated boundary conditions, and orthogonality is a general property of the eigenfunctions. The weight function is one of the coefficients in the differential equation. That's where it comes from. You can read about it at

http://en.wikipedia.org/wiki/Sturm–Liouville_theory