Question: In this problem, we outline a proof of the first part of a theorem which states that the eigenvalues of the Sturm-Liouville problem (1) and (2) are simple (only one independent eigenfunction for each eigenvalue). We will argue by contradiction. For a given \(\lambda\), suppose that \(\phi_{1}\) and \(\phi_{2}\) are two linearly independent eigenfunctions. Compute the Wronskian \(W\left(\phi_{1}, \phi_{2}\right)(x)\) and use the boundary conditions (2) to show that \(W\left(\phi_{1}, \phi_{2}\right)(0)=0\). Then use certain theorems to conclude that \(\phi_{1}\) and \(\phi_{2}\) cannot be linearly independent as assumed.

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