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Applying Initial Conditions

  1. Dec 8, 2004 #1
    I need to find all the separated solns of

    [tex] x^2 \frac{\partial^2 u}{\partial x^2} + x\frac{\partial u}{\partial x} + \frac{\partial^2 u}{\partial y^2} = 0 [/tex]

    in the strip [tex]{(x,y) : 0 < y < a, -\infty < x < \infty } [/tex]
    the separated solns must also satisfy u = 0 on both the edges, that is, on y=0 and y=a for all values of x.

    Iv got the general solutions to be..

    [tex] X(x) = Dlnx + C , (k = 0) [/tex]
    [tex]X(x) = Dx^{n} + Cx^{-n} , (k \neq 0) [/tex]

    and

    [tex]Y(y) = A\cos{ky} + B\sin{ky} , (k \neq 0)[/tex]
    [tex]Y(y) = Ay + B , (k = 0)[/tex]

    where k is just the constant iv let the two bits equal when I separated the variables. (well -k^2 actually).

    I just need help interpreting the conditions to sort out the constants..I think!
     
  2. jcsd
  3. Dec 8, 2004 #2

    Galileo

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    Homework Helper

    I haven't checked your answer, but if it is correct then, since [itex]u(x,y)=X(x)Y(y)[/itex], the boundary conditions say:

    [tex]u(x,0)=X(x)Y(0)=0[/tex]
    and
    [tex]u(x,a)=X(x)Y(a)=0[/tex]

    So [itex]Y(0)=Y(a)=0[/itex]

    For example: if k=0, then applying the boundary condition at y=0 gives:
    [tex]Y(0)=B=0[/tex]
     
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