Consider the functionalS[(y) = R₁²dx ln(1+x²y'), y(1) = 0, y(2) = A,where A is a constant and y is a continuously differentiable function for1 ≤ 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].-2SA = Edx=x²h' €² 21²1 + x²y' 2dydx сwhere c is a nonzero constant.1x²¹dxh(1) = h(2) = 0, then the term O(e) in this expansionvanishes if y'(x) satisfies the equationx4h2(1 + x²y')²+ 0(€³).Solve this equation to show that the stationary path isy(x)x(1+2A) – (3 + 2A) 1+XFor what range of values of A is this derivation valid?

To show that the term in the expansion vanishes under the given conditions, let's consider the…

Consider the functional Slu] = [₁² 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 € be a constant. Let A = S[y+ ch]— S[y]. dx ln(1+x²y'), y(1) = 0, y(2) = A, = E 2 € f² = dx x²h' 1 + x²y' 2 €² 2 = vanishes if y'(x) satisfies the equation 1 x²¹ dx h(1) h(2) = 0, then the term O(e) in this expansion dy dx с where c is a nonzero constant. x4h2 (1 + x²y')² + 0(€³).

Consider the functional Slu] = [₁² 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 € be a constant. Let A = S[y+ ch]— S[y]. dx ln(1+x²y'), y(1) = 0, y(2) = A, = E 2 € f² = dx x²h' 1 + x²y' 2 €² 2 = vanishes if y'(x) satisfies the equation 1 x²¹ dx h(1) h(2) = 0, then the term O(e) in this expansion dy dx с where c is a nonzero constant. x4h2 (1 + x²y')² + 0(€³).

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 14E

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