Existence of a unique solution:

The main motivation for concentrating on problems of class is the following theorem:

Theorem: A boundary value problem of class has a unique solution.

Proof: Let denote the solution of the initial value problem

.

By virtue of a standard theorem in the theory of differential equation (theorem 7.5 of chapter 1 in Coddington and Leuinson), is a continuous function of and , and also exists and is continuous for and all values of.

In order to show that the equation for , has exactly one solution, we shall prove that

(10.18)

The desired result then follows from the fact that a monotone function defined for all values of whose derivative is bounded away from zero assumes every value exactly once.

In order to establish (10.18), we differentiate the identity

with respect to. We get