I need a bit of help with these boundary value problems. I'm trying to find their eigenvalues and eigenfunctions and although I pretty much know how to do it, I want to exactly WHY I'm doing each step. I attached part of my work, and on it I have a little question next to the steps I need explained. The biggest question I have is how do you know what form to choose for the eigenfunctions. Thanks a lot for taking the time to help me out.