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Differential Equation - Uniqueness Theroem

  1. Mar 24, 2009 #1
    1. The problem statement, all variables and given/known data

    The differential equation that models the volume of a raindrop is [tex]\frac{dv}{dt} = kv^{2/3}[/tex] where [tex]k = 3^{2/3}(4 \pi)^{1/3}[/tex]

    A) Why doesn't this equation satisfy the hypothesis of the Uniqueness Theroem?
    B) Give a physical interpertation of the fact that solution to this equation with the initial condition v(0) = 0 are not unique. Does this model say anything about the way raindrops begin to form?

    2. Relevant equations



    3. The attempt at a solution

    A) The equation doesn't satisfy the hypothesis Uniqueness Theroem because when v = 0, the equation's derivative does not exist.

    B) At time t = 0, the raindrop does not have volume, but as t increases, it's volume increases as well.

    Am I correct for both parts
     
    Last edited: Mar 24, 2009
  2. jcsd
  3. Mar 24, 2009 #2

    HallsofIvy

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    Do you mean "dv/dt", rather than "dy/dt"?

    Strictly speaking an "equation" doesn't have a derivative. What you mean is that the function [itex]kv^{2/3}[/itex] has no derivative at v= 0. That is true and is a reason why the uniqueness theorem does not hold.

    How do you conclude that "its volume increases"? Certainly v(t)= 0 for all t satisfies [itex]dv/dt= kv^{2/3}[/itex] as well as v(0)= 0.

     
  4. Mar 24, 2009 #3
    so to prove that the volume does increase, I would need to find it's general solution?
     
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