We have emphasized that the Uniqueness Theorem does not apply to every differential equation. There are hy- potheses that must be verified before we can apply the theorem. However, there is a temptation to think that, since models of "real world" problems must obviously have solutions, we don't need to worry about the hypotheses of the Uniqueness Theorem when we are working with differential equations modeling the physical world. The following models illustrates the flaw in this assumption. Suppose we wish to study the formation of raindrops in the atmosphere. We make the reasonable assumption that raindrops are approximately spherical. We also assume that the rate of growth of the volume of a raindrop is proportional to its surface area. Let r(t) be the radius of the raindrop at time t, S(t) be its surface area at time t, and V (t) be its volume at time t. We know from basic geometry that S = 4rr? and V = (a) Show that the differential equation that models the volume of the raindrop under these assumptions is AP = kV²/3 dt where k is a proportionality constant. (b) Why doesn't this equation satisfy the hypothesis of the Uniqueness Theorem?
We have emphasized that the Uniqueness Theorem does not apply to every differential equation. There are hy- potheses that must be verified before we can apply the theorem. However, there is a temptation to think that, since models of "real world" problems must obviously have solutions, we don't need to worry about the hypotheses of the Uniqueness Theorem when we are working with differential equations modeling the physical world. The following models illustrates the flaw in this assumption. Suppose we wish to study the formation of raindrops in the atmosphere. We make the reasonable assumption that raindrops are approximately spherical. We also assume that the rate of growth of the volume of a raindrop is proportional to its surface area. Let r(t) be the radius of the raindrop at time t, S(t) be its surface area at time t, and V (t) be its volume at time t. We know from basic geometry that S = 4rr? and V = (a) Show that the differential equation that models the volume of the raindrop under these assumptions is AP = kV²/3 dt where k is a proportionality constant. (b) Why doesn't this equation satisfy the hypothesis of the Uniqueness Theorem?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Please do both a and b
Transcribed Image Text:5.
We have emphasized that the Uniqueness Theorem does not apply to every differential equation. There are hy-
potheses that must be verified before we can apply the theorem. However, there is a temptation to think that, since
models of "real world" problems must obviously have solutions, we don't need to worry about the hypotheses of the
Uniqueness Theorem when we are working with differential equations modeling the physical world. The following
models illustrates the flaw in this assumption.
Suppose we wish to study the formation of raindrops in the atmosphere. We make the reasonable assumption that
raindrops are approximately spherical. We also assume that the rate of growth of the volume of a raindrop
is proportional to its surface area.
Let r(t) be the radius of the raindrop at time t, S(t) be its surface area at time t, and V(t) be its volume at time t.
We know from basic geometry that
4
S = 4rr?
V = r".
and
3
(a) Show that the differential equation that models the volume of the raindrop under these assumptions is
dV
= kV²/3
dt
where k is a proportionality constant.
(b) Why doesn't this equation satisfy the hypothesis of the Uniqueness Theorem?
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