If a mod sees this, please delete the topic; I posted it in the wrong section.

[cse63146]

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]

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?

Homework Equations

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

 

[nate808]

?

A) Yes, you are correct. The Uniqueness Theorem requires the equation to be continuously differentiable, meaning that the derivative exists at all points. In this case, the derivative does not exist when v = 0, so the theorem does not apply.

B) Yes, you are correct again. The initial condition v(0) = 0 means that at time t = 0, the raindrop has no volume. However, as time increases, the volume of the raindrop increases due to the velocity term in the equation. This could be interpreted as raindrops forming from smaller droplets as they fall through the atmosphere.

 

[Architeuthis Dux]

?



A) Yes, you are correct. The Uniqueness Theorem states that for a differential equation of the form y' = f(t,y) with initial condition y(t0) = y0, there exists a unique solution in some interval around t0 if f(t,y) is continuous and satisfies the Lipschitz condition. In this case, the equation does not satisfy the Lipschitz condition because the derivative is not defined at v = 0.

B) Your interpretation is correct. The initial condition v(0) = 0 represents the beginning of the raindrop's formation, but as time increases, the volume of the raindrop increases as well. This model does not take into account the formation process of raindrops, but rather focuses on the growth of the raindrop once it has formed.