Consider the functional S[y] = [² da ln(1 + a²y'), 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 € be a constant. Let A = S[y + ch] - S[y]. 2 = cff² dx A = E +0(€³). if h(1) = h(2) = 0, then the term O(e) in this expansion vanishes if y'(x) satisfies the equation = y(x) x²h' €² 2 1+x²y S² = dy 1 1 dx с where c is a nonzero constant. Solve this equation to show that the stationary path is x(1+2A) − (3+2A) 1 X dx 2 x4h2 (1 + x²y')² +-. X
Consider the functional S[y] = [² da ln(1 + a²y'), 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 € be a constant. Let A = S[y + ch] - S[y]. 2 = cff² dx A = E +0(€³). if h(1) = h(2) = 0, then the term O(e) in this expansion vanishes if y'(x) satisfies the equation = y(x) x²h' €² 2 1+x²y S² = dy 1 1 dx с where c is a nonzero constant. Solve this equation to show that the stationary path is x(1+2A) − (3+2A) 1 X dx 2 x4h2 (1 + x²y')² +-. X
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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