Boundary Value Problems

TL;DR: Two-point boundary value problems for ODE are not always solvable, and the theory is more subtle than for initial value problems.

Abstract: The basic theory of boundary value problems for ODE is more subtle than for initial value problems, and we can give only a few highlights of it here. For nota-tional simplicity, abbreviate boundary value problem by BVP. We begin with the two-point BVP y = f (x, y, y), a < x < b A y(a) y (a) + B y(b) y (b) = γ 1 γ 2 with A and B square matrices of order 2. These matrices A and B are given, along with the right-hand boundary values γ 1 and γ 2. The problem may not be solvable (or uniquely solvable) in all cases, and we illustrate this with some examples (also given in the text).