Question 1. (a) Verify that the sequence U„(x) = cos (n + x) n e N form an orthogonal set of functions on [0, 7]. That is, prove Un(x)Um(x)dx = 0, for n + m. (The formula cos(a) cos(b) = 3(cos(a – b) + cos(a + b)) may be useful). (b) 1 Let ø be a C' function, and suppose 00 $(x) = b„Un(x), $"(x) = enUn(x). n=1 n=1 Express the coefficients c, in terms of b, and the values ø(0), ø(1), Ø'(0), Ø'(n).
Question 1. (a) Verify that the sequence U„(x) = cos (n + x) n e N form an orthogonal set of functions on [0, 7]. That is, prove Un(x)Um(x)dx = 0, for n + m. (The formula cos(a) cos(b) = 3(cos(a – b) + cos(a + b)) may be useful). (b) 1 Let ø be a C' function, and suppose 00 $(x) = b„Un(x), $"(x) = enUn(x). n=1 n=1 Express the coefficients c, in terms of b, and the values ø(0), ø(1), Ø'(0), Ø'(n).
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 1. (a)
Verify that the sequence
1
Un(x) = cos (n + ;)x),
n e N
form an orthogonal set of functions on [0,1]. That is, prove
| Un(x)Um(x)dx = 0, for n # m.
(The formula cos(a) cos(b) = ¿(cos(a – b) + cos(a + b)) may be useful).
(b) I
Let ø be a C' function, and suppose
$(x) = > b„Un(x)., ø"(x) = > cnUn(x).
%3D
n=1
n=1
Express the coefficients c, in terms of b, and the values ø(0), ø(7), Ø'(0), ø'(n).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc2397270-c13f-4cbc-8085-ef9e58d3496f%2Fc1e406f3-2d6a-455a-8e3a-f65da5a9bbea%2Fhw6j03o_processed.png&w=3840&q=75)
Transcribed Image Text:Question 1. (a)
Verify that the sequence
1
Un(x) = cos (n + ;)x),
n e N
form an orthogonal set of functions on [0,1]. That is, prove
| Un(x)Um(x)dx = 0, for n # m.
(The formula cos(a) cos(b) = ¿(cos(a – b) + cos(a + b)) may be useful).
(b) I
Let ø be a C' function, and suppose
$(x) = > b„Un(x)., ø"(x) = > cnUn(x).
%3D
n=1
n=1
Express the coefficients c, in terms of b, and the values ø(0), ø(7), Ø'(0), ø'(n).
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