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. functional S[y] = fºd dx f(x)√1+y¹², y(a) = A, y(b) = B, stationary path of the
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. functional S[y] = fºd dx f(x)√1+y¹², y(a) = A, y(b) = B, stationary path of the
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.
functional
S[y] = [° dæƒ(x)√²+y^², y(a) = A, y(b) = B,
a
is given by
x
S [²
a
y(x) = A + B
B-A B
dw
1
√f (w)² - 3²¹
where is a constant satisfying
·b
= 8° de
dw
stationary path of the
1
√f (w)² – ß²*
Using the inequality (which is valid for all real z and u)
zu
√1 + (z+u)² − √√1 + x² >
√1+z²
or otherwise, show that the stationary path gives a
global minimum of the functional S[y].
??](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F59d44c96-efb1-4f3c-83b3-5a6a84cf94cb%2Fc3b82db5-a6dd-41ac-924b-d3ebb6962f7a%2Fqxbn22_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.
functional
S[y] = [° dæƒ(x)√²+y^², y(a) = A, y(b) = B,
a
is given by
x
S [²
a
y(x) = A + B
B-A B
dw
1
√f (w)² - 3²¹
where is a constant satisfying
·b
= 8° de
dw
stationary path of the
1
√f (w)² – ß²*
Using the inequality (which is valid for all real z and u)
zu
√1 + (z+u)² − √√1 + x² >
√1+z²
or otherwise, show that the stationary path gives a
global minimum of the functional S[y].
??
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