I am currently holding a discussion class on diff eqs for engineers and I am looking for an interesting conceptual problem on eigenvalues in diff eqs.
Most of the problems in 5 different books that I have searched in go like this:
Suppose that $y'' + \lambda y = 0$ where $y(0) = 0, y(0) = \pi$. Find the values of $\lambda$ for which a nontrivial solution exists and find the corresponding eigenvector.
While I think that the above routine problem is a good practice, the students have done this a lot in HW.
I am looking for a conceptual problem (geometric thinking would be great too) to test whether students really understand the concept above.
Or is there really nothing much to add?
I'd like to hear from you for any suggestions. Thanks!