I am trying to understand Courant's minimax theorem, and am hoping someone can provide me with a simple worked example showing how to find an approximation to, say, the second or third eigenvalue of a 1- or 2- dimensional differential equation system, ideally using the same notation as in Methods of Mathematical Physics, if possible, as a lot of references I have found on the internet use matrix notation which I am not familiar with. I can post the original statement of the theorem if that helps.
If it is any help I think I follow the Rayleigh-Ritz method to find the lowest eigenvalue, by minimising a trial function, and understand that higher eigenvalues can be found if you know the lower eigenfunctions, but don't understand how to apply Courant's method where you do not need to know the lower eigenfunctions.
Thanks!