My article Boundary Value Problems and Finite Differences has just been appeared in the January 2016 issue of the College Mathematics Journal. Here is the abstract:

The solvability of boundary value problems differs greatly from that of initial value problems and can be somewhat difficult to make sense of in the context of a sophomore-level differential equations course. We present an approach that uses finite difference approximations to motivate and understand the theory governing the existence and uniqueness of solutions to boundary value problems at an elementary level.