Let a < b and let ƒ(x) be a continuously differentiable function on the interval [a, b] with f(x) > 0 for all x = [a, b]. Let A > 0, B> 0 be constants. Show that the stationary path of the functional S[(y)] = [" is given by [° dx f(x)√1+y¹², y(a) = A, y(b) = B, rx 1 y(x) = A + 8 [² du √ √5)² - 32 dw - where is a constant satisfying · A=D [ ² de √(1-3³²-13² dw a B-A= B
Let a < b and let ƒ(x) be a continuously differentiable function on the interval [a, b] with f(x) > 0 for all x = [a, b]. Let A > 0, B> 0 be constants. Show that the stationary path of the functional S[(y)] = [" is given by [° dx f(x)√1+y¹², y(a) = A, y(b) = B, rx 1 y(x) = A + 8 [² du √ √5)² - 32 dw - where is a constant satisfying · A=D [ ² de √(1-3³²-13² dw a B-A= B
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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![Let a < b and let f(x) be a continuously differentiable function on the
interval [a, b] with f(x) > 0 for all x = [a, b].
Let A > 0, B > 0 be constants. Show that the stationary path of the
functional
·b
S[y] ["*
=
is given by
y(x)
dx f(x)√1+y¹², y(a) = A, y(b) = B,
S²
= A+B
dw
1
√f(w)² – ß²¹
where is a constant satisfying
1
B-A=0 [ dw √f (w)
√f (w)² – ß²](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F59d44c96-efb1-4f3c-83b3-5a6a84cf94cb%2Ffe251648-0d9b-4d46-9a4a-49d1b56e235d%2F81nuw0h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let a < b and let f(x) be a continuously differentiable function on the
interval [a, b] with f(x) > 0 for all x = [a, b].
Let A > 0, B > 0 be constants. Show that the stationary path of the
functional
·b
S[y] ["*
=
is given by
y(x)
dx f(x)√1+y¹², y(a) = A, y(b) = B,
S²
= A+B
dw
1
√f(w)² – ß²¹
where is a constant satisfying
1
B-A=0 [ dw √f (w)
√f (w)² – ß²
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