Differential Equations with Applications and Historical Notes

George F. Simmons

• Publisher: Chapman and Hall/CRC
• Year: 2016
• ISBN:
  9781498702621
• Edition: 3
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

Fads are as common in mathematics as in any other human activity, and it is always difficult to separate the enduring from the ephemeral in the achievements of one's own time. An unfortunate effect of the predominance of fads is that if a student doesn't learn about such worthwhile topics as the wave equation, Gauss's hypergeometric function, the gamma function, and the basic problems of the calculus of variations—among others—as an undergraduate, then he/she is unlikely to do so later.

The natural place for an informal acquaintance with such ideas is a leisurely introductory course on differential equations. Specially designed for just such a course, Differential Equations with Applications and Historical Notes takes great pleasure in the journey into the world of differential equations and their wide range of applications. The author—a highly respected educator—advocates a careful approach, using explicit explanation to ensure students fully comprehend the subject matter.

With an emphasis on modeling and applications, the long-awaited Third Edition of this classic textbook presents a substantial new section on Gauss's bell curve and improves coverage of Fourier analysis, numerical methods, and linear algebra. Relating the development of mathematics to human activity—i.e., identifying why and how mathematics is used—the text includes a wealth of unique examples and exercises, as well as the author's distinctive historical notes, throughout.

• 
  
  

• Provides an ideal text for a one- or two-semester introductory course on differential equations

• 
  
  

• Emphasizes modeling and applications

• 
  
  

• Presents a substantial new section on Gauss's bell curve

• 
  
  

• Improves coverage of Fourier analysis, numerical methods, and linear algebra

• 
  
  

• Relates the development of mathematics to human activity—i.e., identifying why and how mathematics is used

• 
  
  

• Includes a wealth of unique examples and exercises, as well as the author's distinctive historical notes, throughout

• 
  
  

• Uses explicit explanation to ensure students fully comprehend the subject matter

Outstanding Academic Title of the Year, Choice magazine, American Library Association.

Chapter 1

The Nature of Differential Equations. Separable Equations

1 Introduction

An equation involving one dependent variable and its derivatives with respect to one or more independent variables is called a differential equation. Many of the general laws of nature—in physics, chemistry, biology, and astronomy—find their most natural expression in the language of differential equations. Applications also abound in mathematics itself, especially in geometry, and in engineering, economics, and many other fields of applied science.

It is easy to understand the reason behind this broad utility of differential equations. The reader will recall that if y = f(x) is a given function, then its derivative dy/dx can be interpreted as the rate of change of y with respect to x. In any natural process, the variables involved and their rates of change are connected with one another by means of the basic scientific principles that govern the process. When this connection is expressed in mathematical symbols, the result is often a differential equation.

The following example may illuminate these remarks. According to Newton’s second law of motion, the acceleration a of a body of mass m is proportional to the total force F acting on it, with 1/m as the constant of proportionality, so that a = F/m or

$ma=F.$

(1)

Suppose, for instance, that a body of mass m falls freely under the influence of gravity alone. In this case the only force acting on it is mg, where g is the acceleration due to gravity.¹ If y is the distance down to the body from some fixed height, then its velocity v = dy/dt is the rate of change of position and its acceleration a = dv/dt = d²y/dt² is the rate of change of velocity. With this notation, (1) becomes

$m\frac{d^{2}y}{d{t}^2}=mg$

or

$\frac{d^{2}y}{d{t}^2}=g.$

(2)

If we alter the situation by assuming that air exerts a resisting force proportional to the velocity, then the total force acting on the body is mg − k(dy/dt), and (1) becomes

$m\frac{d^{2}y}{d{t}^2}=mg-k\frac{dy}{dt}.$

(3)

Equations (2) and (3) are the differential equations that express the essential attributes of the physical processes under consideration.

As further examples of differential equations, we list the following:

$\frac{dy}{dt}=-ky;$

(4)

$m\frac{d^{2}y}{d{t}^2}=-ky;$

(5)

$\frac{dy}{dx}+2xy=e^{-x^2};$

(6)

$\frac{d^{2}y}{d{x}^2}-5\frac{dy}{dx}+6y=0;$

(7)

$(1-x^2)\frac{d^{2}y}{d{x}^2}-2x\frac{dy}{dx}+p(p+1)y=0;$

(8)

$x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+(x^2-p^2)y=0.$

(9)

The dependent variable in each of these equations is y, and the independent variable is either t or x. The letters k, m, and p represent constants. An ordinary differential equation is one in whi...