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# Wave Equations and Periodic Differentiable Functions

3. Solve the wave equation,

&#8706;2u/&#8706;t2 = c2(&#8706;2u/&#8706;x) -&#8734; < x < &#8734;

With initial conditions, u(x,0) = (1/x2+1)sin(x), and &#8706;u/&#8706;t(x,0) = x/(x2+1)

4. Suppose that f is a 2&#1087;-periodic differentiable function with Fouier coefficients a0, an and bn. Consider the Fourier coefficients of f ' given by

a0 = 1/2&#1087;&#8747; f '(x) dx, an = 1/&#1087; &#8747; f '(x) cos(nx) dx, bn = 1/&#1087; &#8747; 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,

&#8706;2u/&#8706;t2 = c2(&#8706;2u/&#8706;x) -&#8734; < x < &#8734;

With initial conditions, u(x,0) = (1/x2+1)sin(x), and &#8706;u/&#8706;t(x,0) = x/(x2+1)

4. Suppose that f is a 2&#1087;-periodic differentiable function with Fouier coefficients a0, an and bn. Consider the Fourier coefficients of f ' given by

a0 = 1/2&#1087;&#8747; f '(x) dx, an = 1/&#1087; &#8747; f '(x) cos(nx) dx, bn = 1/&#1087; &#8747; 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.

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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 ...

#### Solution Summary

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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