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. S[y] = da [* dx f(x)}√/1+ y^², _y(a) = A, y(b) = B, a is given by y(x) = rx [² d A+B B-A= B dw where is a constant satisfying b 35° a 1 √f(w)² - 3² dw 1 √f (w)² — B²*
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. S[y] = da [* dx f(x)}√/1+ y^², _y(a) = A, y(b) = B, a is given by y(x) = rx [² d A+B B-A= B dw where is a constant satisfying b 35° a 1 √f(w)² - 3² dw 1 √f (w)² — B²*
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.
S[y] =
is given by
y(x)
=
[* dx ƒ(x)√1+y¹², _y(a) = A₁ y(b) = B,
A,
a
A+B
1² d
dw
1
√f(w)²- ß²
where is a constant satisfying
b
B-A= B 3 [º
a
dw
1
√f (w)² - 82
the stationary path
gives a weak local minimum of the functional S[y].
Using the Jacobi equation,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F59d44c96-efb1-4f3c-83b3-5a6a84cf94cb%2Fac95d0b0-8f36-475d-b02f-a94d863bf29e%2Fctlkkq_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.
S[y] =
is given by
y(x)
=
[* dx ƒ(x)√1+y¹², _y(a) = A₁ y(b) = B,
A,
a
A+B
1² d
dw
1
√f(w)²- ß²
where is a constant satisfying
b
B-A= B 3 [º
a
dw
1
√f (w)² - 82
the stationary path
gives a weak local minimum of the functional S[y].
Using the Jacobi equation,
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