Consider the functional S[y] 2 = = 1₁² dx ln(1 + x²y'), 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 e be a constant. Let A = S[y+ ch] - S[y]. 2 = - de - dez dx S dx = x²h' €² 1 + x²y' 2 vanishes if y'(x) satisfies the equation y(x) = dy 1 1 dx с where c is a nonzero constant. if h(1) = h(2) = 0, then the term O(e) in this expansion x2, x(1+2A) (3 + 2A) 2 x4h2 (1 + x²y)² + the stationary path is 1 X +0(€3³). .
Consider the functional S[y] 2 = = 1₁² dx ln(1 + x²y'), 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 e be a constant. Let A = S[y+ ch] - S[y]. 2 = - de - dez dx S dx = x²h' €² 1 + x²y' 2 vanishes if y'(x) satisfies the equation y(x) = dy 1 1 dx с where c is a nonzero constant. if h(1) = h(2) = 0, then the term O(e) in this expansion x2, x(1+2A) (3 + 2A) 2 x4h2 (1 + x²y)² + the stationary path is 1 X +0(€3³). .
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
By examining the value of the O(ϵ2) term in ∆, determine whether S[y] has a local maximum or minimum on the stationary path.
![Consider the functional
2
S[(y) = ₁² dx ln(1 + x²y'), 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 e be a constant. Let A = S[y+ ch] - S[y].
2
x²h'
2
x = ef de + ² y = = ² ² + 1/² +0(e).
E dx
dx
+0(€³).
1 x²y' 2
(1+x²y')²
if h(1) = h(2) = 0, then the term O(e) in this expansion
vanishes if y'(x) satisfies the equation
dy
dx C
where c is a nonzero constant.
-
y(x) =
=
1
x²¹
x(1 + 2A) − (3 + 2A)
2
the stationary path is
1
-
X
+](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F59d44c96-efb1-4f3c-83b3-5a6a84cf94cb%2F192487ff-76bf-4cdf-8c8b-2b9451c3cd3e%2Fk2r407xq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the functional
2
S[(y) = ₁² dx ln(1 + x²y'), 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 e be a constant. Let A = S[y+ ch] - S[y].
2
x²h'
2
x = ef de + ² y = = ² ² + 1/² +0(e).
E dx
dx
+0(€³).
1 x²y' 2
(1+x²y')²
if h(1) = h(2) = 0, then the term O(e) in this expansion
vanishes if y'(x) satisfies the equation
dy
dx C
where c is a nonzero constant.
-
y(x) =
=
1
x²¹
x(1 + 2A) − (3 + 2A)
2
the stationary path is
1
-
X
+
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