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Prove that the eigenfunctions corresponding to different eigenvalues (of the following
Chapter 5, Problem 5.5.2(choose chapter or problem)
Prove that the eigenfunctions corresponding to different eigenvalues (of the following eigenvalue problem) are orthogonal:
\(\frac{d}{d x}\left[p(x) \frac{d \phi}{d x}\right]+q(x) \phi+\lambda \sigma(x) \phi=0\)
with the boundary conditions
\(\begin{aligned}
\phi(1) & =0 \\
\phi(2)-2 \frac{d \phi}{d x}(2) & =0 .
\end{aligned}\)
What is the weighting function?
Questions & Answers
QUESTION:
Prove that the eigenfunctions corresponding to different eigenvalues (of the following eigenvalue problem) are orthogonal:
\(\frac{d}{d x}\left[p(x) \frac{d \phi}{d x}\right]+q(x) \phi+\lambda \sigma(x) \phi=0\)
with the boundary conditions
\(\begin{aligned}
\phi(1) & =0 \\
\phi(2)-2 \frac{d \phi}{d x}(2) & =0 .
\end{aligned}\)
What is the weighting function?
ANSWER:Step 1 of 3
Let \(\phi_{1}\) and \(\phi_{2}\) be the eigenfunction of the Sturm-Liouville operator L corresponding to two different eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\)