Consider the functional S[y] 2 = = 1₁² dx ln(1 + x²y'), y(1) = 0, y(2) = A, where A is a constant and y is a continuously differentiable function for 1 ≤ x ≤ 2. Let h be a continuously differentiable function for 1 ≤ x ≤ 2, and let e be a constant. Let A = S[y+ ch] - S[y]. 2 = - de - dez dx S dx = x²h' €² 1 + x²y' 2 vanishes if y'(x) satisfies the equation y(x) = dy 1 1 dx с where c is a nonzero constant. if h(1) = h(2) = 0, then the term O(e) in this expansion x2, x(1+2A) (3 + 2A) 2 x4h2 (1 + x²y)² + the stationary path is 1 X +0(€3³). .

Consider the functional S[y] 2 = = 1₁² dx ln(1 + x²y'), y(1) = 0, y(2) = A, where A is a constant and y is a continuously differentiable function for 1 ≤ x ≤ 2. Let h be a continuously differentiable function for 1 ≤ x ≤ 2, and let e be a constant. Let A = S[y+ ch] - S[y]. 2 = - de - dez dx S dx = x²h' €² 1 + x²y' 2 vanishes if y'(x) satisfies the equation y(x) = dy 1 1 dx с where c is a nonzero constant. if h(1) = h(2) = 0, then the term O(e) in this expansion x2, x(1+2A) (3 + 2A) 2 x4h2 (1 + x²y)² + the stationary path is 1 X +0(€3³). .

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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Question

By examining the value of the O(ϵ2) term in ∆, determine whether S[y] has a local maximum or minimum on the stationary path.

Consider the functional
2
S[(y) = ₁² dx ln(1 + x²y'), y(1) = 0, y(2) = A,
where A is a constant and y is a continuously differentiable function for
1 ≤ x ≤ 2. Let h be a continuously differentiable function for 1 ≤ x ≤ 2,
and let e be a constant. Let A = S[y+ ch] - S[y].
2
x²h'
2
x = ef de + ² y = = ² ² + 1/² +0(e).
E dx
dx
+0(€³).
1 x²y' 2
(1+x²y')²
if h(1) = h(2) = 0, then the term O(e) in this expansion
vanishes if y'(x) satisfies the equation
dy
dx C
where c is a nonzero constant.
-
y(x) =
=
1
x²¹
x(1 + 2A) − (3 + 2A)
2
the stationary path is
1
-
X
+

Formula Formula A function f(x) attains a local maximum at x=a , if there exists a neighborhood (a−δ,a+δ) of a such that, f(x)<f(a), ∀ x∈(a−δ,a+δ),x≠a f(x)−f(a)<0, ∀ x∈(a−δ,a+δ),x≠a In such case, f(a) attains a local maximum value f(x) at x=a .

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Step 1: Conceptual Introduction

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Step 2: Expanding the Functional S[y+εh] Using Taylor Series

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Step 3: Integrating the Expanded Terms Separately

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Step 4: Calculating the Functional S[y]

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Step 5: Finding the Difference Δ

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Determine S[y] has a local maximum or minimum on the stationary path. (by the value of the O(ϵ2) term in Δ)

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