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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). .

Calculus For The Life Sciences
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...
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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 +
Transcribed Image Text: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 +
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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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