(b) Show that if h(1) = h(2) = 0, then the term O(e) in this expansion vanishes if y'(x) satisfies the equation dy dx с where c is a nonzero constant. 1 1
(b) Show that if h(1) = h(2) = 0, then the term O(e) in this expansion vanishes if y'(x) satisfies the equation dy dx с where c is a nonzero constant. 1 1
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
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![Consider the functional
S[y] = [² da ln(1+z²y), y(1) = 0, v(2) = A,
dx
where A is a constant and y is a continuously differentiable function for
1 ≤ x ≤ 2. Let h be a continuously differentiable function for 1 ≤ x ≤ 2,
and let € be a constant. Let A = S[y+ ch] – S[y].
A = €
x²h'
dx
1 1 + x²y'
dy 1 1
dx
с
€² 2
+0(€³).
(b) Show that if h(1) = h(2) = 0, then the term O(e) in this expansion
vanishes if y'(x) satisfies the equation
2'
X²
where c is a nonzero constant.
dx
x4h2
(1 + x²y')²](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F59d44c96-efb1-4f3c-83b3-5a6a84cf94cb%2F706815a7-9481-4cb0-8a1c-649b4f913a97%2Fexdl2rn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the functional
S[y] = [² da ln(1+z²y), y(1) = 0, v(2) = A,
dx
where A is a constant and y is a continuously differentiable function for
1 ≤ x ≤ 2. Let h be a continuously differentiable function for 1 ≤ x ≤ 2,
and let € be a constant. Let A = S[y+ ch] – S[y].
A = €
x²h'
dx
1 1 + x²y'
dy 1 1
dx
с
€² 2
+0(€³).
(b) Show that if h(1) = h(2) = 0, then the term O(e) in this expansion
vanishes if y'(x) satisfies the equation
2'
X²
where c is a nonzero constant.
dx
x4h2
(1 + x²y')²
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