Consider the boundary value problem: d»(x) = k²>(x); v(x) = b(x+ L) dx?
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
a-c step by step please
![1. Consider the boundary value problem:
dv(x)
= k?»(x); v(x) = p(x+ L)
(1)
%3D
dx?
with x E [a, b] and L = b – a.
a) Show that the problem admits solutions of the form (x) = Ceika, where C is a
constant.
b) Find the corresponding eigenvalues and eigenfunctions.
c) Show that the eigenfunctions are orthogonal. Find C, such that the orthogonality
relation reads:
V (x)Vn(x)dx = 8mn
d) These eigenfunctions constitute a complete orthonormal set {Þn(x)}-∞ for x E
[a, b], which it means that any piecewise periodic function, f (x) = f(x + L) can be
expressed as:
f (x) = anvn(x)
n=-0
Find an expression for an.
e) Show that the orthogonal expansion of the function, f(x), shown below can be
expressed as
n+1
f (")-Σ .
- sin(nAx)
n=1
1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f6d2f58-ebc5-487b-abb4-91cd16bfc1c1%2F2c466ca6-4b8b-4641-8035-66462aa11c72%2Fwxwin9r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Consider the boundary value problem:
dv(x)
= k?»(x); v(x) = p(x+ L)
(1)
%3D
dx?
with x E [a, b] and L = b – a.
a) Show that the problem admits solutions of the form (x) = Ceika, where C is a
constant.
b) Find the corresponding eigenvalues and eigenfunctions.
c) Show that the eigenfunctions are orthogonal. Find C, such that the orthogonality
relation reads:
V (x)Vn(x)dx = 8mn
d) These eigenfunctions constitute a complete orthonormal set {Þn(x)}-∞ for x E
[a, b], which it means that any piecewise periodic function, f (x) = f(x + L) can be
expressed as:
f (x) = anvn(x)
n=-0
Find an expression for an.
e) Show that the orthogonal expansion of the function, f(x), shown below can be
expressed as
n+1
f (")-Σ .
- sin(nAx)
n=1
1
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