3. Solve the wave equation,
∂2u/∂t2 = c2(∂2u/∂x) -∞ < x < ∞
With initial conditions, u(x,0) = (1/x2+1)sin(x), and ∂u/∂t(x,0) =
x/(x2+1)
4. Suppose that f is a 2п-periodic differentiable function with Fouier
coefficients a0, an and bn. Consider the Fourier coefficients of f '
given by
a0 = 1/2п∫ f '(x) dx, an = 1/п ∫ f '(x) cos(nx) dx, bn =
1/п ∫ f '(x) sin(nx) dx,
a) Show that a0 = 0.
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coefficients of f ' in terms of the Fourier coefficients of f.
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Mathematics Homework Solutions
#167122
Wave Equations and Periodic Differentiable Functions
3. Solve the wave equation,
∂2u/∂t2 = c2(∂2u/∂x) -∞ < x < ∞
With initial conditions, u(x,0) = (1/x2+1)sin(x), and ∂u/∂t(x,0) = x/(x2+1)
4. Suppose that f is a 2п-periodic differentiable function with Fouier coefficients a0, an and bn. Consider the Fourier coefficients of f ' given by
a0 = 1/2п∫ f '(x) dx, an = 1/п ∫ f '(x) cos(nx) dx, bn = 1/п ∫ f '(x) sin(nx) dx,
a) Show that a0 = 0.
b) Using integration by parts on the formula for an and bn, find a formula for the Fourier coefficients of f ' in terms of the Fourier coefficients of f.
3. Solve the wave equation,
∂2u/∂t2 = c2(∂2u/∂x) -∞ < x < ∞
With initial conditions, u(x,0) = (1/x2+1)sin(x), and ∂u/∂t(x,0) = x/(x2+1)
4. Suppose that f is a 2п-periodic differentiable function with Fouier coefficients a0, an and bn. Consider the Fourier coefficients of f ' given by
a0 = 1/2п∫ f '(x) dx, an = 1/п ∫ f '(x) cos(nx) dx, bn = 1/п ∫ f '(x) sin(nx) dx,
a) Show that a0 = 0.
b) Using integration by parts on the formula for an and bn, find a formula for the Fourier coefficients of f ' in terms of the Fourier coefficients of f.
Wave Equations and Periodic Differentiable Functions are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.
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