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Find the equilibrium solution for an autonomous equation.

  1. Aug 2, 2011 #1
    1. The problem statement, all variables and given/known data

    Consider a cylindrical water tank of constant cross section A. Water is pumped into the tank at a constant rate k and leaks out through a small hole of area a in the bottom of the tank. From Torcelli’s principle in hydrodynamics, it follows that the rate at which water flows through the hole is (alpha)(a)squareroot((2)(g)(h)) , where h is the current depth of water in the tank, g is the acceleration due to gravity, and alpha is a contraction coefficient that satisfies 0.5 < alpha < 1.0.

    1. Show that the depth of water in the tank at any time satisfies the equation
    dh/dt = ([k] - [(alpha)(a)squareroot{(2)(g)(h)}])/A
    2. Determine the equilibrium depth he , of water, and show that it is asymptotically stable. Observe that he does not depend on A.

    2. Relevant equations

    We covered how to solve a DE using the integrating factor method after putting the eq. in to standard form but now we have moved on to autonomous eq.'s and I'm a little unsure how to go about solving this.

    The problem looks more complex than it is because of all the brackets I had to use but I think it is pretty straight forward for someone familiar with the subject.

    3. The attempt at a solution


    dh/dt = rate in - rate out

    = (volume in/min)(1/area) - (volume out/min)(1/area)
    = [k(m3/min) / A(m2)] - [((alpha)(a)squareroot{(2)(g)(h)}(m3/min) / A(m2)]
    = ([k] - [(alpha)(a)squareroot{(2)(g)(h)}])/A

    (Seems straight forward have I done this wrong?)


    dh/dt = m(h) h

    => [(k/hA - (alpha)(a)squareroot{(2)(g)}/sqrt{h}A] h

    (Skipped the intermediate steps. Is this the right form? What do I do now?)

    Thanks in advance :)
    Last edited: Aug 2, 2011
  2. jcsd
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