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Trying to solve a partial differential equation using d'Alembert's solution

  1. Mar 11, 2009 #1
    Hi, I'm trying to understand this.

    The given equation is y_tt = 4 y_xx
    0 < x < pi, t>0
    where y_tt is the 2nd derivative with respect to t, y_xx is 2nd wrt x

    Boundary conditions
    y(0,t) = 0 and y(pi,t) = 0

    And initial conditions
    y_t (x,0) = 0 = g(x)
    y(x,0) = sin^2 x = f(x)

    ---

    My teacher wrote that F(x) is the odd periodic extension of f(x), and then wrote

    F(x) = sign(sinx)sin^2 x

    1) I assume this is to make it odd but why wouldn't he just write sign(x)sin^2 x?

    2) Also, there was a similar question in class but -infinity < x < infinity and no boundary conditions given with one of the initial conditions y(x,0) = 1/(1+x^2) = f(x). In that case, since f(x) is even why isn't it necessary to use the sign function?

    Thanks for your help!
     
  2. jcsd
  3. Mar 11, 2009 #2

    Hurkyl

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    Staff Emeritus
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    Gold Member

    Oddness wasn't the only property he wanted.
     
  4. Mar 11, 2009 #3
    Why does the sin^2 need to be corrected like that?
     
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