Partial DIfferential Equations / Eigenvalues

[DualCortex]

Hi, I'm barely a high school senior who is somewhat overwhelmed by a univ. course.
Anyway, we are just learning to solve some basic PDEs using the method of separation of variables.
With this method (and the questions we are given) we check three cases to find the eigenvalues of Sturm-Liouville problems ( which come out from the PDEs): when lambda is > 0, < 0, or = 0.
Up to know, I don't think we have seen any sample problem that has eigenvalues that apply to more than one of the cases. This is what I want to know, can an S-L problem have eigenvalues that span across those three cases?
If so, I'm guessing that if I check one of the cases and it does have eigenvalues that lead to a non-trivial solution, then I can safely ignore the other cases? Thank you so much for your time in advance.

 

[coomast]

Hello DualCortex, welcome to the forum,

You have to check all cases. If you don't do this you might forget part of the solution and the end result will certainly be incorrect. Consider p.e. the following:

https://www.physicsforums.com/showthread.php?t=214251

Here it is important to study all cases. There are two of them which have non-trivial solutions and you need all of them for obtaining the solution using the boundary on the right hand side of the domain. In this particular example you will not be able to find one coefficient because the boundary conditions are all derivatives.

Hope this helps.