1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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:


    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.


    [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.


    [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


    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Autonomous differential equation
  1. Differential equations (Replies: 5)

  2. Differential Equations (Replies: 1)

  3. Differential Equations (Replies: 4)