Differential Equations - Eigenvalues and Eigenfunctions

[solomar]

Homework Statement

Find the eigenvalues and the eigenfunctions for
x^2y"+2xy'+λy = 0 y(1) = 0, y(e^2) = 0

Homework Equations

See problem

The Attempt at a Solution

My book has one paragraph on this that does not help me. I tried using an auxiliary equation and solving for lambda.
I get (m+√λ)^2 a real number with duplicity two. I don't know what to do with this or if this is even the right first step. The SINGLE paragraph on this doesn't explain anything (it's practically a side note)

I would appreciate some guidance. There are two problems like this on the take home quiz and we didn't go over it in class so I am very lost and confused.
Also I've been pounding my head over this with my directionless fury. Help subside my hate!

Thanks for your time everyone!

 

[HallsofIvy]

You mean "multiplicity" 2 ("duplicity" has a completely different meaning!). And, while you say you "get (m+√λ)^2" you don't say what that is supposed to be- particularly since there is no "m" in the problem. Please show how you got that.

(I'm not sure what you mean by "m" but I get nothing like that. It looks to me like $\lambda$ will have to be larger than 1/4 in order to get a non-trivial function that is 0 for two different x values.)

 

[solomar]

I got it from changing the DE into
m^2 + 2m + λ
I did the "m" thing because in the previous chapters to solve some equations I would solve the associated homogeneous DE and use the value of m to determine the structure of the general solution.
It's the only step I have made and I feel like even that is wrong. I don't know what to do with the initial values or anything...
Thanks for clearing up the duplicity multiplicity thing, you're right :)

My problem is a complete lack of understanding of what I am supposed to do, and my textbook is not helping haha...