Problem 1. Let Ao be the collection of admissible functions on [a,b], i.e., Ao = {v : [a, b] → R | v(a) = 0 = v(b)} Suppose that uЄC+1 ([a, b]). Prove that u(x) is a polynomial of degree at most n if and only if for all vЄ A₁ we have d" u v'(x)dx = 0. dan Hint: Make sure you prove both directions of this statement.
Problem 1. Let Ao be the collection of admissible functions on [a,b], i.e., Ao = {v : [a, b] → R | v(a) = 0 = v(b)} Suppose that uЄC+1 ([a, b]). Prove that u(x) is a polynomial of degree at most n if and only if for all vЄ A₁ we have d" u v'(x)dx = 0. dan Hint: Make sure you prove both directions of this statement.
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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![Problem 1.
Let Ao be the collection of admissible functions on [a,b], i.e.,
A0 = {v : [a, b] → R | v(a) = 0 = v(b)}
Suppose that u € C+1 ([a, b]). Prove that u(x) is a polynomial of degree at most n if and only if for all
v € A₁ we have
d" u
v'(x)dx
= 0.
dan
Hint: Make sure you prove both directions of this statement.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d49a1bf-def0-472f-94db-660c9693a140%2F31d14e30-076a-458d-b0fc-f6cd67724217%2Fvgkfbph_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 1.
Let Ao be the collection of admissible functions on [a,b], i.e.,
A0 = {v : [a, b] → R | v(a) = 0 = v(b)}
Suppose that u € C+1 ([a, b]). Prove that u(x) is a polynomial of degree at most n if and only if for all
v € A₁ we have
d" u
v'(x)dx
= 0.
dan
Hint: Make sure you prove both directions of this statement.
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