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Differential Equation - Uniqueness Theroem

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1. Homework Statement

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. Homework 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:

HallsofIvy

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1. Homework Statement

The differential equation that models the volume of a raindrop is [tex]\frac{dy}{dt} = kv^{2/3}[/tex] where [tex]k = 3^{2/3}(4 \pi)^{1/3}[/tex]
Do you mean "dv/dt", rather than "dy/dt"?

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. Homework 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.
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.

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

Am I correct for both parts
 
so to prove that the volume does increase, I would need to find it's general solution?
 

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