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-2", e¬4²). Define a inner product for f, g E W as (f,g) = | f(x)g(x)dx i. For a general real constant k > 0, evaluate the following integral to show that 1 -ka dx e 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-",e-2", e¬4²). Define a inner product for f, g E W as (f,g) = | f(x)g(x)dx i. For a general real constant k > 0, evaluate the following integral to show that 1 -ka dx e 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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please help me solve i and ii, thank you.
Transcribed Image Text: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-2", e¬4²). Define a inner
product for f, g E W as
(f,g)
= | f(x)g(x)dx
i. For a general real constant k > 0, evaluate the following integral to show that
1
-ka dx
e
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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