Consider the functional S[y] = [² da ln(1 + a²y'), y(1) = 0, y(2) = A, 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]. 2 = cff² dx A = E +0(€³). if h(1) = h(2) = 0, then the term O(e) in this expansion vanishes if y'(x) satisfies the equation = y(x) x²h' €² 2 1+x²y S² = dy 1 1 dx с where c is a nonzero constant. Solve this equation to show that the stationary path is x(1+2A) − (3+2A) 1 X dx 2 x4h2 (1 + x²y')² +-. X
Consider the functional S[y] = [² da ln(1 + a²y'), y(1) = 0, y(2) = A, 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]. 2 = cff² dx A = E +0(€³). if h(1) = h(2) = 0, then the term O(e) in this expansion vanishes if y'(x) satisfies the equation = y(x) x²h' €² 2 1+x²y S² = dy 1 1 dx с where c is a nonzero constant. Solve this equation to show that the stationary path is x(1+2A) − (3+2A) 1 X dx 2 x4h2 (1 + x²y')² +-. X
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter9: Multivariable Calculus
Section9.2: Partial Derivatives
Problem 48E
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Question
![Consider the functional
S[y] = [² dx ln(1 + x² y'), y(1) = 0, y(2) = A,
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].
2
= e f₁² da
A = E
1 1
x²h'
1 + x²y'
dy
dx с
where c is a nonzero constant.
y(x)
²f²
1
+0(€³).
if h(1) = h(2) = 0, then the term O(e) in this expansion
vanishes if y' (x) satisfies the equation
=
dx
Solve this equation to show that the stationary path is
x(1+2A) − (3 + 2A) 1
+
2
x4h12
(1 + x²y')²
X](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F59d44c96-efb1-4f3c-83b3-5a6a84cf94cb%2F5e37eebf-b985-41f0-a216-e6bca7ef9743%2Flfb9ls_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the functional
S[y] = [² dx ln(1 + x² y'), y(1) = 0, y(2) = A,
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].
2
= e f₁² da
A = E
1 1
x²h'
1 + x²y'
dy
dx с
where c is a nonzero constant.
y(x)
²f²
1
+0(€³).
if h(1) = h(2) = 0, then the term O(e) in this expansion
vanishes if y' (x) satisfies the equation
=
dx
Solve this equation to show that the stationary path is
x(1+2A) − (3 + 2A) 1
+
2
x4h12
(1 + x²y')²
X
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