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Help with a relatively simple first order, first rank DE

  1. Feb 12, 2009 #1
    Here is the problem. Skip the first two paragraphs to get to the pure math part.

    There are two cubes of water, both with area of 2 x 2 dm^2. Via the bottom they are connected by a tube, and the flow of water is (surprise, surprise) proportional to the difference in water level between the two, by 0.1 * delta h. In addition to that, a constant amount of 1.0 liter / min is pumped from one to the other. At t=0, the pump is turned on when h1 = h2 = 10 dm. Derive a formula that describes delta h as function of t.

    I started with: Dv1 / dt = A * dh1 / dt = 0.1*delta h - 1.0, and a similar equation for h2.
    Substracting these two [i know this is risky, but i didn't know what else to do], i get:
    A* d(h1-h2) / dt = A * d(delta h)/dt = 0.2*delta h - 2.0. Substracting A gives:

    The math part:

    d(delta h) / dt - 0.05 delta h = 0.5

    Let's make things easier by renaming delta h to "x":

    dx/dt - 0.05x = 0.5

    I'm using a standard solution-formula, which says that for:
    dx/dt + a*x = F(t), the solution is:

    x(t) = exp(-a*t) * integrand (F(t')*exp(a*t')) + c*exp(-a*t)

    Using this formula, i get: x(t) = -10 + c* exp(0.05*t).

    Obviously, this problem would lead to a negative exponential as solution, or the difference in water level would go to infinity as time goes by!

    Where do i miss a minus sign?
  2. jcsd
  3. Feb 12, 2009 #2


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    I skipped straight to the math part and found no problems, so I think the problem is in the physics (or, more accurately, the mathematical model of the physical situation) after all.

    You should check all your signs in the equation you are writing down. In particular pay attention to the sign in front of the 0.1 * delta h. Clearly define what delta h is (is it always the height of #1 - the height of #2, so h1 - h2, for example) and convince yourself that if h1 > h2 then your differential equation ensures that delta h is positive (and if h1 < h2 then delta h is negative).
  4. Feb 12, 2009 #3
    Thanks CompuChip, i just checked it and the error is indeed in the derivation of the DE, indeed the minus sign of the 0.1 * delta h.
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