I remember some of my linear algebra from my studies but can't wrap my head around this one.(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

Say my solution to a DE is "f(x)" (happens to be bessel's equation), and it contains a constant variable "d" in the argument of the bessel's functions (i.,e. J(d*x) and Y(d*x)). So my solution is:

f(x)= A*J(d*x) + B*Y(d*x)

I can find a general solution for f(x) by imposing 2 boundary conditions f(x1)=f(x2)=0. That would give me an equation for f_n(x).

First question:The author calls this f_n(x) the "eigenfunctions" and the "orthogonal basis". Why is this given these names? I'm not sure why these solutions form an orthogonal basis.

Second question:

The author then states that an arbitrary vector F(x) "can be expanded in this orthogonal basis" via:

F(x)= sum{from n=1 to inf} [ a_n*f_n(x) ]

where

a_n = [ (f_n(x) , F(x)) ] / [ (f_n(x) , f_n(x) ]

What in the world is this on about? Any guidance would be helpful!

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# Homework Help: Bessel equation & Orthogonal Basis

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