Consider the functional S[y] = ay(1)² + [* dx ßy², y(0) = 0, with a natural boundary condition at x C{y] = Yy(1)² + [" dx w(x) y² = 1, where a, ẞ and y are nonzero constants. = 1 and subject to the constraint Show that the stationary paths of this system satisfy the Euler-Lagrange equation d²y β +\w(x) y = 0, y(0) = 0, (a− yλ) y(1) + ßy′(1) = 0, dx2 where is a Lagrange multiplier.
Consider the functional S[y] = ay(1)² + [* dx ßy², y(0) = 0, with a natural boundary condition at x C{y] = Yy(1)² + [" dx w(x) y² = 1, where a, ẞ and y are nonzero constants. = 1 and subject to the constraint Show that the stationary paths of this system satisfy the Euler-Lagrange equation d²y β +\w(x) y = 0, y(0) = 0, (a− yλ) y(1) + ßy′(1) = 0, dx2 where is a Lagrange multiplier.
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] = ay(1)² + [* dx ßy², y(0) = 0,
with a natural boundary condition at x
C{y] = Yy(1)² + ["
dx w(x) y² = 1,
where a, ẞ and y are nonzero constants.
=
1 and subject to the constraint
Show that the stationary paths of this system satisfy the
Euler-Lagrange equation
d²y
β +\w(x) y = 0, y(0) = 0, (a− yλ) y(1) + ßy′(1) = 0,
dx2
where is a Lagrange multiplier.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa6c8ed7d-75cc-4e27-869e-3ad6a1efc0b4%2Fa6de6ec4-4013-4114-955b-7ffb38ad9b85%2F6vnrt23_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the functional
S[y] = ay(1)² + [* dx ßy², y(0) = 0,
with a natural boundary condition at x
C{y] = Yy(1)² + ["
dx w(x) y² = 1,
where a, ẞ and y are nonzero constants.
=
1 and subject to the constraint
Show that the stationary paths of this system satisfy the
Euler-Lagrange equation
d²y
β +\w(x) y = 0, y(0) = 0, (a− yλ) y(1) + ßy′(1) = 0,
dx2
where is a Lagrange multiplier.
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