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Curious About this DE

  1. Jan 1, 2014 #1
    Hi, lately I've been messing around a lot with the Laplacian operator and DE's including the Laplacian operator. Most recently, the equation below is the one I have been messing around with and trying to understand better.

    [itex]\nabla^2 U(\vec{r})=C(\vec{r})U(\vec{r})[/itex]

    This is pretty general though.. WAYY too general for me to tackle. So I've been starting with the 1D case, which I also can't seem to solve.

    [itex]\frac{d^2}{dx^2}U(x)=C(x)U(x)[/itex]

    My goal is to try to solve for U(x) in terms of C(x). Any ideas? Is there any way to know if such a solution exists? What about to the general equation above?

    Thanks :)
     
    Last edited: Jan 1, 2014
  2. jcsd
  3. Jan 1, 2014 #2

    HallsofIvy

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    Do you know the form of the Laplacian operator in spherical or polar coordinates
     
  4. Jan 1, 2014 #3
    No I don't but it wouldn't be too much of a hassle to figure it out. How could that help though?
     
  5. Jan 8, 2014 #4
    A couple of thoughts for progressing.

    1) Try the case where C is constant. This actually gives you a Helmholtz equation.

    2) For the more general case, it helps to assume that U or U dot n =0 at the boundary and C has a certain sign.
    Then multiply by U and integrate over the domain, this will involve an integration by parts.
     
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