Solved: In this problem we outline a proof of the first part of Theorem 11.2.3: that the

Chapter 11, Problem 20

(choose chapter or problem)

In this problem we outline a proof of the first part of Theorem 11.2.3: that the eigenvalues of the SturmLiouville problem (1), (2) are simple. The proof is by contradiction. (a) Suppose that a given eigenvalue is not simple. Then there exist two corresponding eigenfunctions 1 and 2 that are linearly independent, that is, not multiples of each other. (b) Compute the Wronskian W(1, 2)(x) and use the boundary conditions (2) to show that W(1, 2)(0) = 0. (c) Use Theorem 3.2.6 to reach a contradiction, which establishes that the eigenvalues must be simple, as asserted in Theorem 11.2.3.