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## Homework Statement

I'm given a standard form of Bessel's equation, namely

[tex] x^2y\prime\prime + xy\prime + (\lambda x^2-\nu^2)y = 0 [/tex]

with [itex] \nu = \frac{1}{3} [/itex] and [itex] \lambda [/itex] some unknown constant, and asked to find its eigenvalues and eigenfunctions.

The initial conditions are [itex] y(0)=0 [/itex] and [itex] y\prime (\pi)=0 [/itex].

## The Attempt at a Solution

This is a single question assignment, so it's supposed to be reasonably extensive. What troubles me is that as far as I know, with this being THE typical Bessel equation, aren't the eigenfunctions pretty much a given? (i.e. they will be one sine and one cosine function of [itex] \sqrt{\lambda} [/itex] and x, right?)

***Check my understanding please; the eigenfunctions are the functions in the general solution, and the eigenvalues are their respective coefficients? For example if some simple equation has the solution [tex] y(x) = C_1 e^{ikx} + C_2 e^{-ikx} [/tex] then the eigenfunctions are [itex] e^{\pm ikx} [/itex] and the eigenvalues are [itex] C_1, C_2 [/itex], right? ***

I mean I can derive them, but it's not particularly difficult, especially since we did it in class, which makes me wonder if it's really what the professor wants.

The eigenvalues I should be able to determine with the two initial conditions I'm given, I think. So mainly, I'm curious if I'm doing the right thing, or if I'm completely off base with my interpretation of the question.

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