Exercise 17 Show that the above results can be written independently for the sine and cosine functions to give X+ °x√ To •Hotd Xo cos(kmx) cos(knx) dx = Czo+λ { sin(kmx) sin(knx) dx = { cos(kmx) sin(knx) dx = 0, X mn 8mn 0 m, n‡0 m=n=0 m, n‡0 m=n=0 " (19) (20) (21) where kn = 2nT/λ. [In the last case, one has to deal with a degeneracy of eigenvalues when m = n. Both the cosine and sine functions have the same eigenvalue, although they are linearly independent.]
Exercise 17 Show that the above results can be written independently for the sine and cosine functions to give X+ °x√ To •Hotd Xo cos(kmx) cos(knx) dx = Czo+λ { sin(kmx) sin(knx) dx = { cos(kmx) sin(knx) dx = 0, X mn 8mn 0 m, n‡0 m=n=0 m, n‡0 m=n=0 " (19) (20) (21) where kn = 2nT/λ. [In the last case, one has to deal with a degeneracy of eigenvalues when m = n. Both the cosine and sine functions have the same eigenvalue, although they are linearly independent.]
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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![Exercise 17 Show that the above results can be written independently for the sine and cosine
functions to give
X+ °x√
To
•Hotd
Xo
rzo+λ
cos(kmx) cos(knx) dx =
{
cos(kmx) sin(knx) dx = 0,
sin(kmx) sin(knx) dx =
X
mn
8mn
0
m, n‡0
m=n=0
m, n‡0
m=n=0
"
(19)
(20)
(21)
where kn = 2nT/λ. [In the last case, one has to deal with a degeneracy of eigenvalues when
m = n. Both the cosine and sine functions have the same eigenvalue, although they are linearly
independent.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb8e0ad34-47bd-4edd-8e64-7fbb4be3cef6%2F5c7ded2c-f5a7-4bbf-a340-e0c1a4692ae7%2Fnzde8h_processed.png&w=3840&q=75)
Transcribed Image Text:Exercise 17 Show that the above results can be written independently for the sine and cosine
functions to give
X+ °x√
To
•Hotd
Xo
rzo+λ
cos(kmx) cos(knx) dx =
{
cos(kmx) sin(knx) dx = 0,
sin(kmx) sin(knx) dx =
X
mn
8mn
0
m, n‡0
m=n=0
m, n‡0
m=n=0
"
(19)
(20)
(21)
where kn = 2nT/λ. [In the last case, one has to deal with a degeneracy of eigenvalues when
m = n. Both the cosine and sine functions have the same eigenvalue, although they are linearly
independent.]
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