Consider the functional Slu] = [₁² 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]. dx ln(1+x²y'), y(1) = 0, y(2) = A, = E 2 € f² = dx x²h' 1 + x²y' 2 €² 2 = vanishes if y'(x) satisfies the equation 1 x²¹ dx h(1) h(2) = 0, then the term O(e) in this expansion dy dx с where c is a nonzero constant. x4h2 (1 + x²y')² + 0(€³).
Consider the functional Slu] = [₁² 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]. dx ln(1+x²y'), y(1) = 0, y(2) = A, = E 2 € f² = dx x²h' 1 + x²y' 2 €² 2 = vanishes if y'(x) satisfies the equation 1 x²¹ dx h(1) h(2) = 0, then the term O(e) in this expansion dy dx с where c is a nonzero constant. x4h2 (1 + x²y')² + 0(€³).
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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![Consider the functional
S[(y) = R₁²
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
S
A = E
dx
=
x²h' €² 2
1²
1 + x²y' 2
dy
dx с
where c is a nonzero constant.
1
x²¹
dx
h(1) = h(2) = 0, then the term O(e) in this expansion
vanishes if y'(x) satisfies the equation
x4h2
(1 + x²y')²
+ 0(€³).
Solve this equation to show that the stationary path is
y(x)
x(1+2A) – (3 + 2A) 1
+
X
For what range of values of A is this derivation valid?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F59d44c96-efb1-4f3c-83b3-5a6a84cf94cb%2F91672368-6e5e-4b4a-a218-82fde013103e%2F05v3qgs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the functional
S[(y) = R₁²
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
S
A = E
dx
=
x²h' €² 2
1²
1 + x²y' 2
dy
dx с
where c is a nonzero constant.
1
x²¹
dx
h(1) = h(2) = 0, then the term O(e) in this expansion
vanishes if y'(x) satisfies the equation
x4h2
(1 + x²y')²
+ 0(€³).
Solve this equation to show that the stationary path is
y(x)
x(1+2A) – (3 + 2A) 1
+
X
For what range of values of A is this derivation valid?
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