e*, when 20 = -1 %3D P„(x) sin(nrx) – nn cos(nax), when 2, = (nx)² for n = 1,2,3,.... a.) Show that Pp • Pq J. ,r)@,(x)dx = 0 %3D for p + q and compute this dot product when p = q.
e*, when 20 = -1 %3D P„(x) sin(nrx) – nn cos(nax), when 2, = (nx)² for n = 1,2,3,.... a.) Show that Pp • Pq J. ,r)@,(x)dx = 0 %3D for p + q and compute this dot product when p = q.
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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Question
![**Orthogonal (Perpendicular) Functions**
You had seen in Problem #2 of Homework #3 that the non-zero solutions \(\varphi(x)\) to the ODE
\[
\varphi''(x) + \lambda \varphi(x) = 0, \, \text{for } 0 < x < 1,
\]
along also the BCs \(\varphi(0) + \varphi'(0) = 0\) and \(\varphi(1) + \varphi'(1) = 0\), are
\[
\varphi_n(x) =
\begin{cases}
e^{-x}, & \text{when } \lambda_0 = -1 \\
\sin(n \pi x) - n \pi \cos(n \pi x), & \text{when } \lambda_n = (n \pi)^2 \, \text{for } n = 1, 2, 3, \ldots
\end{cases}
\]
a.) Show that
\[
\varphi_p \cdot \varphi_q = \int_0^1 \varphi_p(x) \varphi_q(x) dx = 0
\]
for \(p \neq q\) and compute this dot product when \(p = q\).
b.) Determine the coefficients \(A_n\) if the function \(f(x) = 1\) is to be expanded as
\[
f(x) = 1 = \sum_{n=0}^{\infty} A_n \varphi_n(x) \quad \text{for all } \quad 0 < x < 1,
\]
where the \(\varphi_n(x)\)'s (for \(n = 0, 1, 2, 3, \ldots\)) are the functions above.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe5d528f6-f704-4253-8470-44f2fbc0de86%2F05fa21b8-c966-43d3-b56d-0b6efd722724%2F1ss0bdh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Orthogonal (Perpendicular) Functions**
You had seen in Problem #2 of Homework #3 that the non-zero solutions \(\varphi(x)\) to the ODE
\[
\varphi''(x) + \lambda \varphi(x) = 0, \, \text{for } 0 < x < 1,
\]
along also the BCs \(\varphi(0) + \varphi'(0) = 0\) and \(\varphi(1) + \varphi'(1) = 0\), are
\[
\varphi_n(x) =
\begin{cases}
e^{-x}, & \text{when } \lambda_0 = -1 \\
\sin(n \pi x) - n \pi \cos(n \pi x), & \text{when } \lambda_n = (n \pi)^2 \, \text{for } n = 1, 2, 3, \ldots
\end{cases}
\]
a.) Show that
\[
\varphi_p \cdot \varphi_q = \int_0^1 \varphi_p(x) \varphi_q(x) dx = 0
\]
for \(p \neq q\) and compute this dot product when \(p = q\).
b.) Determine the coefficients \(A_n\) if the function \(f(x) = 1\) is to be expanded as
\[
f(x) = 1 = \sum_{n=0}^{\infty} A_n \varphi_n(x) \quad \text{for all } \quad 0 < x < 1,
\]
where the \(\varphi_n(x)\)'s (for \(n = 0, 1, 2, 3, \ldots\)) are the functions above.
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