Consider the functional S[y] = ay(1)² + √² dx By, y(0) = 0, == with a natural boundary condition at x = 1 and subject to the constraint C[y] = √y(1)² + [* dx w(x) y² = 1, where a, ẞ and y are nonzero constants. Show that the stationary paths of this system satisfy the Euler-Lagrange equation d²y B +Aw(x) y = 0, y(0) = 0, (a-yλ) y(1) + ßy' (1) = 0, dx² where A is a Lagrange multiplier.

Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter14: Discrete Dynamical Systems
Section14.3: Determining Stability
Problem 13E: Repeat the instruction of Exercise 11 for the function. f(x)=x3+x For part d, use i. a1=0.1 ii...
Question
Consider the functional
S[y] = ay(1)² + √²
dx By, y(0) = 0,
==
with a natural boundary condition at x = 1 and subject to the constraint
C[y] = √y(1)² + [* dx w(x) y² = 1,
where a, ẞ and y are nonzero constants.
Show that the stationary paths of this system satisfy the
Euler-Lagrange equation
d²y
B +Aw(x) y = 0, y(0) = 0, (a-yλ) y(1) + ßy' (1) = 0,
dx²
where A is a Lagrange multiplier.
Transcribed Image Text:Consider the functional S[y] = ay(1)² + √² dx By, y(0) = 0, == with a natural boundary condition at x = 1 and subject to the constraint C[y] = √y(1)² + [* dx w(x) y² = 1, where a, ẞ and y are nonzero constants. Show that the stationary paths of this system satisfy the Euler-Lagrange equation d²y B +Aw(x) y = 0, y(0) = 0, (a-yλ) y(1) + ßy' (1) = 0, dx² where A is a Lagrange multiplier.
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