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Homework Help: Autonomous differential equation

  1. Jul 4, 2005 #1
    I just need some help with this problem. Thank you.

    "A pond forms as water collects in a conical depression of radius [tex]a[/tex] and depth [tex]h[/tex]. Suppose that water flows in at a constant rate [tex]k[/tex] and is lost through evaporation at a rate proportional to the surface area.

    (a) Show that the volume [tex]V(t)[/tex] of water in the pond at time [tex]t[/tex] satisfies the differential equation

    [tex]\frac{dV}{dt} = k - \alpha \pi \left( \frac{3a}{\pi h} \right) ^{2/3} V ^{2/3}[/tex]

    where [tex]\alpha[/tex] is the coefficient of evaporation.

    (b) Find the equilibrium depth of water in the pond. Is the equilibrium asymptotically stable?

    (c) Find a condition that must be satisfied if the pond is not to overflow."

    My work:

    (a)

    Consider the following

    [tex]\frac{dV}{dt} = k - \alpha \pi \underbrace{\left( \frac{3a}{\pi h} \right) ^{2/3} V ^{2/3}} _{r^2}[/tex]

    where [tex]r[/tex] is the radius of the pond. Thus, we have

    [tex]r^2 = \left( \frac{3a}{\pi h} \right) ^{2/3} V ^{2/3}[/tex]

    [tex]r^3 = \frac{3aV}{\pi h}[/tex]

    [tex]V = \frac{1}{3} \pi r^2 \left( \frac{hr}{a} \right) = \frac{1}{3} \pi r^2 L[/tex]

    which satisfies the differential equation.

    (b)

    [tex]\frac{dV}{dt} = 0[/tex]

    [tex]k - \alpha \pi r^2 = 0[/tex]

    [tex]r = \sqrt{\frac{k}{\alpha \pi}}[/tex]

    Then, the equilibrium depth of water in the pond is

    [tex]L = \frac{h}{a} \sqrt{\frac{k}{\alpha \pi}}[/tex]

    In this particular case, I don't know how to show whether or not it is asymptotically stable.

    (c)

    [tex]L = \frac{h}{a} \sqrt{\frac{k}{\alpha \pi}} \leq h[/tex]


    Any help is highly appreciated.
     
  2. jcsd
  3. Jul 5, 2005 #2
    An equilibrium depth of water in the pond implies that there is an equilibrium volume. Thus, a direction field can show whether or not it is asymptotically stable, which is what I have at

    http://mygraph.cjb.net/

    Here are the (random) values that I used to plot it:

    [tex]\left\{ \begin{array}{ll} k = 1 \\ \alpha = 0.6 \\ a = 0.3 \\ h = 0.5 \end{array} \right.[/tex]

    Based on this information, I'd say the equilibrium is asymptotically stable.

    Thanks anyway
     
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