Chapter 1.5, Problem 18E

We have emphasized that the Uniqueness Theorem does not apply to every differential equation. There are hypotheses 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 model 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 arc approximately spherical. We also assume that the rate of growth of the volume of a raindrop is proportional to its surf ace area.
Let r(t) be the radius of the raindrop at timet, s(t) be its surface area at time t, and v(t) be its volume at time t. From three-dimensional geometry, we know that
   and .
(a) Show that the differential equation that models the volume of the raindrop und er these assumptions is
  
where k is a proportionality constant.
(b) Why doesn’t this equation satisfy the hypotheses of the Uniqueness Theorem?
(c) Give a physical interpretation of the fact that solutions 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?