FirstOrder Differential Equations and Their （一阶微分方程及其）.pdf

CHAPTER 1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● First-Order Differential Equations and Their Applications 1.1 Introduction to Ordinary Differential Equations Differential equations are found in many areas of mathematics, science, and engineer- ing. Students taking a first course in differential equations have often already seen simple examples in their mathematics, physics, chemistry, or engineering courses. If you have not already seen differential equations, go to the library or Web and glance at some books or journals in your major field. You may be surprised to see the way in which differential equations dominate the study of many aspects of science and engineering. Applied mathematics involves the relationships between mathematics and its applications. Often the type of mathematics that arises in applications is differential equations. Thus, the study of differential equations is an integral part of applied math- ematics. Applied mathematics is said to have three fundamental aspects, and this course will involve a balance of the three: 1. The modeling process by which physical objects and processes are described by physical laws and mathematical formulations. Since so many physical laws involve rates of change (or the derivative), differential equations are often the natural language of science and engineering. 2. The analysis of the mathematical problems that are posed. This involves the complete investigation of the differential equation and its solutions, including detailed numerical studies. We will say more about this shortly. 3. However, the mathematical solution of the differential equation does not complete the overa