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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
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.
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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