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Math Help - Wave equations

  1. #1
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    Wave equations

    First, please forgive the poor notation used here as LaTeX does not seem to be working properly for the time being (http://www.mathhelpforum.com/math-help/f47/working-latex-issues-178319-2.html)?

    This is a problem from "Elementary Differential Equations and Boundary Value Problems" by Boyce and DiPrima. Problem 10.7.1, one that should be trivial, I assume.

    "Consider an elastic string of length L whose ends are held fixed. The string is set in motion with no initial velocity from an initial position u(x, 0) = f(x). Find the displacement u(x, t) for the given initial position f(x)."

    f(x) = (2x)/L when 0 <= 0 <= L/2
    f(x) = (2(L-x))/L when L/2 < x <= L

    I eventually arrive at c_n = (2/L) * definite integral from 0 to L of (f(x) * sin((n*pi*x)/L)) dx
    c_n = (2/L) * definite integral from 0 to L/2 of (((2x)/L * sin((n*pi*x)/L)) dx + (2/L) * definite integral from L/2 to L of ((2(L-x)/L) * sin((n*pi*x)/L)) dx)

    How should I go about evaluating these integrals? I believe it has something to do with odd and even functions?

    From the Student Solutions Manual I see they arrive at the same c_n prior to evaluating the integrals, and after they have evaluated them (this step is not elaborated at all) they arrive at:

    c_n = (8/(n^2 * pi^2)*sin((n*pi)/2)
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  2. #2
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    Your setup is correct.

    You need to use integration by parts.

    \frac{2}{L}\int_{0}^{\frac{L}{2}} \frac{2x}{L}\sin\left( \frac{n \pi x}{L}\right)dx=\frac{4}{L^2}\int_{0}^{\frac{L}{2}} x\sin\left( \frac{n \pi x}{L}\right)dx

    u=x \implies du=dx \quad dv=\sin\left( \frac{n \pi x}{L}\right) \implies v=-\left( \frac{L}{n \pi}\right)\cos\left( \frac{n \pi x}{L}\right)

    \frac{4}{L^2}\left( -x\cdot\left( \frac{L}{n \pi}\right)\cos\left( \frac{n \pi x}{L}\right) \right)\bigg|_{0}^{\frac{L}{2}}+\int_{0}^{\frac{L}  {2}}\left( \frac{L}{n \pi}\right)\cos\left( \frac{n \pi x}{L}\right)dx=

    Now do the same thing to the other integral and simplify.
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