Consider the eigenvalue problem X"(x) - pX(x) = 0 (0 < x < a) X'(0) = 0, X'(a) = -3X(a), where p is a constant. Show that if A> 0 and if p = -12 is an eigenvalue, then 3 cot(a) = A.
Consider the eigenvalue problem X"(x) - pX(x) = 0 (0 < x < a) X'(0) = 0, X'(a) = -3X(a), where p is a constant. Show that if A> 0 and if p = -12 is an eigenvalue, then 3 cot(a) = A.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Eigenvalue Problem
Consider the eigenvalue problem
\[
\begin{cases}
X''(x) - pX(x) = 0 & (0 < x < a) \\
X'(0) = 0, & X'(a) = -3X(a),
\end{cases}
\]
where \( p \) is a constant.
### Objective
Show that if \( \lambda > 0 \) and if \( p = -\lambda^2 \) is an eigenvalue, then \( 3 \cot (\lambda a) = \lambda \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa68164dd-6bba-4aa5-92bc-4824a71db092%2F4ed4c42f-0e2a-4ee5-9707-df1d9554cd02%2Fs1v8oyk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Eigenvalue Problem
Consider the eigenvalue problem
\[
\begin{cases}
X''(x) - pX(x) = 0 & (0 < x < a) \\
X'(0) = 0, & X'(a) = -3X(a),
\end{cases}
\]
where \( p \) is a constant.
### Objective
Show that if \( \lambda > 0 \) and if \( p = -\lambda^2 \) is an eigenvalue, then \( 3 \cot (\lambda a) = \lambda \).
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