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 ·b S[y] = [* dx ƒ(x)√¹+ y^², y(a) = A, v(b) = B, a is given by y(x) = A + B X [² a 1 dw √f(w)² – 3² 9 where is a constant satisfying B-A= B ·b = 3 [ dre- √5(201²-3² dw stationary path of the Using the inequality (which is valid for all real z and u) zu /1 + (z+u)² − √√/1 + x² > √1+z²¹ or otherwise, show that the stationary path gives a global minimum of the functional S[y]. 2 9
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 ·b S[y] = [* dx ƒ(x)√¹+ y^², y(a) = A, v(b) = B, a is given by y(x) = A + B X [² a 1 dw √f(w)² – 3² 9 where is a constant satisfying B-A= B ·b = 3 [ dre- √5(201²-3² dw stationary path of the Using the inequality (which is valid for all real z and u) zu /1 + (z+u)² − √√/1 + x² > √1+z²¹ or otherwise, show that the stationary path gives a global minimum of the functional S[y]. 2 9
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] =
is given by
a
dx f(x)√1+y¹², y(a) = A, y(b) = B,
y (x) = A +8 √ ²
dw
where is a constant satisfying
B-A-B dw
a
1
9
√f(w)² – B²
1
√f(W)
√f(w)² - B²
●
stationary path of the
Using the inequality (which is valid for all real z and u)
zu
√1+(z+u)² = √1+z²>
√1+z²
9
or otherwise, show that the stationary path gives a
global minimum of the functional S[y].
2
9](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F59d44c96-efb1-4f3c-83b3-5a6a84cf94cb%2F345eed94-649c-4371-afa8-793d56290582%2Ff95f5xt_processed.png&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] =
is given by
a
dx f(x)√1+y¹², y(a) = A, y(b) = B,
y (x) = A +8 √ ²
dw
where is a constant satisfying
B-A-B dw
a
1
9
√f(w)² – B²
1
√f(W)
√f(w)² - B²
●
stationary path of the
Using the inequality (which is valid for all real z and u)
zu
√1+(z+u)² = √1+z²>
√1+z²
9
or otherwise, show that the stationary path gives a
global minimum of the functional S[y].
2
9
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