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Consider the vector space of solutions W to the differential equation y(3) + 7y(2) + 14y/ + 8y = 0 We can assume without proof that a basis for W is B = (e-a, e-2", e-4«). Define a inner product for f, g E W as (f,9) = f(2)g(x)dæ i. For a general real constant k > 0, evaluate the following integral to show that 1 k ii. Apply the Gram-Schmidt algorithm on B (in the given order) to find an orthogonal basis B = (h1, h2, h3) for W.

Consider the vector space of solutions W to the differential equation y(3) + 7y(2) + 14y/ + 8y = 0 We can assume without proof that a basis for W is B = (e-a, e-2", e-4«). Define a inner product for f, g E W as (f,9) = f(2)g(x)dæ i. For a general real constant k > 0, evaluate the following integral to show that 1 k ii. Apply the Gram-Schmidt algorithm on B (in the given order) to find an orthogonal basis B = (h1, h2, h3) for W.

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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Evaluate the following integral and apply the Gram-Schmidt algorithm on B to find an orthogonal basis for W. If can answer both (i)and(ii), else answer (ii)

(b) Consider the vector space of solutions W to the differential equation y(3). + 7y(2) +14y' + 8y = 0 We can assume without proof that a basis for W is B = (e-, e product for f, gEW as ", e-4«). Define a inner (f, 9) = | f(x)g(x)dx i. For a general real constant k > 0, evaluate the following integral to show that 1 -k* dx k ii. Apply the Gram-Schmidt algorithm on B (in the given order) to find an orthogonal basis B = (h1, h2, h3) for W.
Transcribed Image Text:(b) Consider the vector space of solutions W to the differential equation y(3). + 7y(2) +14y' + 8y = 0 We can assume without proof that a basis for W is B = (e-, e product for f, gEW as ", e-4«). Define a inner (f, 9) = | f(x)g(x)dx i. For a general real constant k > 0, evaluate the following integral to show that 1 -k* dx k ii. Apply the Gram-Schmidt algorithm on B (in the given order) to find an orthogonal basis B = (h1, h2, h3) for W.
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