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Poisson's equation in 1D

  1. Sep 27, 2009 #1
    1. The problem statement, all variables and given/known data
    Alright, well this is more of a math problem I guess... but here it is.
    [tex]\frac{\partial^{2}}{\partial x{2}} V = - \frac{I}{A\epsilon_o} \sqrt{\frac{m}{2e}} \frac{1}{\sqrt{V}} [/tex]

    Everything besides the [tex]V[/tex] is constant.

    2. Relevant equations
    Trying to solve for the potential


    3. The attempt at a solution
    What's the general solution for these types of second order differentials?
     
  2. jcsd
  3. Sep 28, 2009 #2

    gabbagabbahey

    User Avatar
    Homework Helper
    Gold Member

    First, if it is a 1D problem, the partial derivatives can be replaced by ordinary derivatives!

    Second, try defining a new variable: [itex]W\equiv \frac{dV}{dx}[/itex] and then take note of the fact that

    [tex]\frac{d^2 V}{dx^2}=\frac{dW}{dx}=\frac{dV}{dx}\frac{dW}{dV}=W\frac{dW}{dV}=\frac{1}{2}\frac{d}{dV}(W^2)[/itex]

    ...you will then have a first order, separable ODE for [itex]W[/itex]....which I'm sure you know how to solve!:wink:

    P.S. This is a common trick, so it's worth remembering!:smile:
     
    Last edited: Sep 28, 2009
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