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Differential equation help

  1. Feb 12, 2016 #1
    I understand what is in the picture http://postimg.org/image/u5ib33kzb/
    but the book goes on to say that the solution is thus of the form
    ## X_n = a_n sin \frac{n \pi x}{l} ##
    How does putting ##β=σ^2=\frac{n^2π^2}{l^2}## into (6.37) result in that?
     
  2. jcsd
  3. Feb 12, 2016 #2

    RUber

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    I apologize in advance if I am off base on this, I cannot access your link.

    I will assume that you have a differential equation that looks like:
    ##x'' +\beta x = 0 ##
    with boundary conditions:
    ##x(0)=x(l) = 0##
    The general solution for the differential equation is
    ##x = A \sin( \sqrt{\beta} t ) + B \cos(\sqrt{\beta} t) ##
    And the boundary condition at ##t=0## forces B to go to zero and the boundary condition at ##t = l ## forces ##\beta ## to be the form you have above.

    Please include a little more information regarding the problem if you would like more feedback.
     
  4. Feb 12, 2016 #3
    thats right, the differential equation is (X is X(x), a function of x) :

    ##X'' + \beta X = 0##

    Assuming a general solution of ##X(x) = A e^{ -\sqrt{-\beta}x} + B e^{+\sqrt{-\beta} x} ##, that ##\sqrt{-\beta}## is complex (ie. ##\beta =σ^2##) , and that the boundary conditions are ##X(0)=0## and ##X(l)=0##, they found that ##σ=\frac{n\pi}{l}##
    What I want to know is how putting ##σ=\frac{n\pi}{l}## into the general solution results in ##X_n=a_n\sin\frac{n\pi x}{l}##
    Or how their general solution is equivalent to yours?
     
    Last edited: Feb 12, 2016
  5. Feb 12, 2016 #4

    pasmith

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    [tex]\cos x = \frac{e^{ix} + e^{-ix}}2 \\
    \sin x = \frac{e^{ix} - e^{-ix}}{2i}[/tex]
     
  6. Feb 12, 2016 #5

    RUber

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    Applying your first boundary condition tells you that A = -B, giving ## X(x) = A\left(e^{-i\sigma x}- e^{i\sigma x}\right)##
    Noting what pasmith wrote above, this is equivalent to ## C \sin (\sigma x )##.
    Then, since any sigma of the form given can be a solution, your full solution might be an infinite sum:
    ##X(x) =\sum_{n=1}^\infty X_n(x) = \sum_{n=1}^\infty a_n \sin(\sigma_n x ) ##
     
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