In Chapter 2, we encountered the awkwardly named concept of differential-equation-with-initial-value problem as the key to our study of natural population growth. Indeed, this concept is key to the study of many problems of growth and decay, of motion, of all sorts of change. In this chapter, we shorten the name to initial value problem, and we focus on this fundamental concept.

After reviewing definitions and elementary examples, we will study two more problems that can be modeled as initial value problems:

• the change in temperature of a body cooling in the air — such as a cup of coffee — and
  
  
• the speed of an object falling under the force of gravity — such as a skydiver.

These two problems are prototypes for all the differential equation problems we will encounter in this course. That is, at the most basic level, every differential equation we see will be like one or the other of these problems.

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