In this problem we outline a proof of the first part of

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 eigenvaluesof 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 correspondingeigenfunctions 1 and 2 that are linearly independentthat is, not multiples of eachother.(b) Compute the Wronskian W(1, 2)(x), and use the boundary conditions (2) to showthat W(1, 2)(0) = 0.(c) Use Theorem 3.2.7 to reach a contradiction, which establishes that the eigenvaluesmust be simple, as asserted in Theorem 11.2.3.