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Diff Equations: Wave Equation

  1. May 11, 2009 #1
    1. The problem statement, all variables and given/known data

    Find a formal solution to the vibrating string problem..

    alpha=4, 0<x<pi t>0
    u(0,t)=u(pi,t)=0 t>0

    f(x)= x^2(pi-x)
    g(x)=0

    2. Relevant equations

    u(x,t) = sum[a cos(alpha*n*t/L + b sin(alpha*n*t/L)*sin(n pi x / L)

    Fourier series for sine

    3. The attempt at a solution

    a = 2/pi * integral(x^2 (pi-x) * sin(nx) dx) from 0 to pi
    = [-2((n pi)^2 - 2)(-1)^n - 4 + 2((n pi)^2 - 6)(-1)^n] / n^3 ... by breaking it into addition of 2 integrals and using the tabular method of integration

    b = 2/pi * integral(0 * sin(nx) dx) from 0 to pi
    = 0


    however, the answer is
    u(x,t) = sum[ 4/n^3 * [2(-1)^(n+1) - 1] * cos2nt sinnx


    Did I do something wrong or is the answer in a different form? I don't see how to get from my answer to the correct one.
     
  2. jcsd
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