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(c) By evaluating the Fourier series for appropriate values of x, find the following sums: (-1)"+1 n² + 1 (i) X= - Σ n=1 1 n² + 1 and (ii) Y = Σ Y= n=1 (d) Find the corresponding Fourier coefficients in the Fourier series for y(x), which is periodic and satisfies the differential equation d²y - y = f(x). dx²

(c) By evaluating the Fourier series for appropriate values of x, find the following sums: (-1)"+1 n² + 1 (i) X= - Σ n=1 1 n² + 1 and (ii) Y = Σ Y= n=1 (d) Find the corresponding Fourier coefficients in the Fourier series for y(x), which is periodic and satisfies the differential equation d²y - y = f(x). dx²

Advanced Engineering Mathematics
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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3. Let the function f be defined by for - < x < π and satisfy f(x+2) = f(x). Recall that cosh x = (e* + e*)/2 and in the following you are given the indefinite integral So 1 a² + b² for constants a and b. cosh(ax) cos(bx)dx= f(x) = cosh x ∞ (i) x = Σ X n=1 (c) By evaluating the Fourier series for appropriate values of x, find the following sums: (a sinh(ax) cos(bx) + b cosh(ax) sin(bx)) 1 n²+1 8 and (ii) Y = Σ n=1 (-1)n+1 n² + 1 (d) Find the corresponding Fourier coefficients in the Fourier series for y(x), which is periodic and satisfies the differential equation d²y dx² - y = f(x).
Transcribed Image Text:3. Let the function f be defined by for - < x < π and satisfy f(x+2) = f(x). Recall that cosh x = (e* + e*)/2 and in the following you are given the indefinite integral So 1 a² + b² for constants a and b. cosh(ax) cos(bx)dx= f(x) = cosh x ∞ (i) x = Σ X n=1 (c) By evaluating the Fourier series for appropriate values of x, find the following sums: (a sinh(ax) cos(bx) + b cosh(ax) sin(bx)) 1 n²+1 8 and (ii) Y = Σ n=1 (-1)n+1 n² + 1 (d) Find the corresponding Fourier coefficients in the Fourier series for y(x), which is periodic and satisfies the differential equation d²y dx² - y = f(x).
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