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We felt it desirable to give the reader a brief but nontrivial introduction to this important topic.
- Differential equations and their applications : an introduction to applied mathematics
- Differential Equations and Their Applications
- Differential equations and their applications solutions manual
Differential equations and their applications : an introduction to applied mathematics
Section 4. In Section 4. Richardson's theory of the escalation of arms races and fit his model to the arms race which led eventually to World War I. This section also provides the reader with a concrete feeling for the concept of stability. The theory we develop here also has a spectacular application to the spraying of insecticides.
In section 4. This model enables us to prove the famous "threshold theorem of epidemiology," which states that an epidemic will occur only if the number of people susceptible to the disease exceeds a certain threshold value. We also compare the predictions of our model with data from an actual plague in Bombay. This textbook also contains the following important, and often unique features.
In Section 1. Our proof is based on the method of Picard iterates, and can be fully understood by anyone who has completed one year of calculus. This section has the added advantage of reinforcing the reader's understanding of the proof of the existence-uniqueness theorem. Complete Fortran and APL programs are given for every computer example in the text.
Computer problems appear in Sections 1. Using this appendix we have been able to teach our students APL in just two lectures. Modesty aside, Section 2. We are very proud of this section because it eliminates all the ambiguities which are inherent in the traditional exposition of this topic.
All the linear algebra pertinent to the study of systems of equations is presented in Sections 3. One advantage of our approach is that the reader gets a concrete feeling for the very important but extremely abstract properties of linear independence, spanning. Indeed, many linear algebra students sit in on our course to find out what's really going on in their course. Differential Equations and Their Applications can he used for a one-or two-semester course in ordinary differential equations.
It is geared to the student who has completed two semesters of calculus. Traditionally, most authors present a "suggested syllabus" for their textbook. We will not do so here, though, since there are already more than twenty different syllabi in use. Suffice it to say that this text can be used for a wide variety of courses in ordinary differential equations. I greatly appreciate the help of the following people in the preparation of this manuscript: Douglas Reber who wrote the Fortran programs, Eleanor Addison who drew the original figures, and Kate MacDougall, Sandra Spinacci, and Miriam Green who typed portions of this manuscript.
I am grateful to Walter Kaufmann-Biihler, the mathematics editor at Springer-Verlag, and Elizabeth Kaplan, the production editor, for their extensive assistance and courtesy during-the preparation of this manuscript. It is a pleasure to work with these true professionals. Finally, I am especially grateful to Joseph P.
LaSalle for the enwuragement and help he gave me. Thanks again, Joe. IntroductionThis book is a study of differential equations and their applications.
A differential equation is a relationship between a function of time and its derivatives. The equations and ii are both examples of differential equations. The order of a differential equation is the order of the highest derivative of the function y that appears in the equation.
Thus i is a first-order differential equation and ii is a third-order differential equation. By a solution of a differential equation we will mean a continuous function y t which together with its derivatives satisfies the relationship.
In this book, we will present serious discussions of the applications of differential equations to such diverse and fascinating problems as the detection of art forgeries, the diagnosis of diabetes, the increase in the percentage of sharks present in the Mediterranean Sea during World War I, and the spread of gonorrhea. Our purpose is to show how researchers have used differential equations to solve, or try to solve, real life problems.
And while we will discuss some of the great success stories of differential equations, we will also point out their limitations and document some of their failures. We approach this problem in the following manner.
A fundamental principle of mathematics is that the way to solve a new problem is to reduce it, in some manner, to a problem that we have already solved. In practice this usually entails successively simplifying the problem until it resembles one we have already solved. Since we are presently in the business of solving differential equations, it is advisable for us to take inventory and list all the differential equations we can solve.
To solve Equation 2 simply integrate both sides with respect to t, which yields Here c is an arbitrary constant of integration, and by ig t dt we mean an anti-derivative of g, that is, a function whose derivative is g. Thus, to solve any other differential equation we must somehow reduce it to the form 2.
As we will see in Section 1. Hence, we will not be able, without the aid of a computer, to solve most differential equations. It stands to reason, therefore, that to find those differential equations that we can solve, we should start with very simple equations Fortunately, the homogeneous equation 4 can be solved quite easily.
Hence Equation 4 can be written in the form But this is Equation 2 "essentially" since we can integrate both sides of 5 to obtain that where c, is an arbitrary constant of integration. But if the absolute value of a continuous function g t is constant then g itself must be constant. Hence, we obtain the equation y t exp or Equation 7 is said to be the general solution of the homogeneous equation since every solution of 4 must be of this form.
Observe that an arbitrary constant c appears in 7. This should not be too surprising. Indeed, we will always expect an arbitrary constant to appear in the general solution of any first-order differential equation. Observe also that Equation 4 has infinitely many solutions; for each value of c we obtain a distinct solution y t.
Rather, we are looking for the specific solution y t which at some initial time to has the value yo. Thus, we want to determine a function y t such that Equation 8 is referred to as an initial-value problem for the obvious reason that of the totality of all solutions of the differential equation, we are looking for the one solution which initially at time to has the value yo. To find this solution we integrate both sides of 5 between to and t.
Thus and, therefore Taking exponentials of both sides of this equation we obtain thatThe function inside the absolute value sign is a continuous function of time. To determine which one it is, evaluate it at the point to; since we see that Hence 1 First-order differential equations Solution. However, this solution is equally as valid and equally as useful as the solution to Example 3. The reason for this is twofold. First, there are very simple numerical schemes to evaluate the above integral to any degree of accuracy with the aid of a computer.
Second, even though the solution to Example 3 is given explicitly, we still cannot evaluate it at any time t without the aid of a table of trigonometric functions and some sort of calculating aid, such as a slide rule, electronic calculator or digital computer.
More precisely, we can multiply both sides of 3 by any continuous function y t to obtain the equivalent equation 1. But this is a first-order linear homogeneous equation for p t , i. Notice how we used our knowledge of the solution of the homogeneous equation to find the function p t which enables us to solve the nonhomogeneous equation. This is an excellent illustration of how we use our knowledge of the solution of a simpler problem to solve a harder problem.
Multiplying both sides of the equation by p t we obtain that 1. In each of Problems , find the solution of the given initial-value problem. When we derived the solution of the nonhomogeneous equation we tacitly assumed that the functions a? Problems illustrate the variety of things that other waste products in a material called slag.
Thus, most of the supply of lead-2 10 is cut off and it begins to decay very rapidly, with a half-life of 22 years. This process continues until the lead in the white lead is once more in radioactive equilibrium with the small amount of radium present, i. Let us now use this information to compute the amount of lead present in a sample in terms of the amount originally present at the time of manufacture.
Let y t be the amount of lead per gram of white lead at time t, yo the amount of lead per gram of white lead present at the time of manufacture to, and r t the number of disintegrations of radium per minute per gram of white lead, at time t.
If A is the decay constant for lead-2 10, thenSince we are only interested in a time period of at most years we may assume that the radium, whose half-life is years, remains constant, so that r t is a constant r. Thus, if we knew yo we could use Equation 5 to compute tto and consequently, we could determine the age of the painting.
As we pointed out, though, we cannot measure yo directly. One possible way out of this difficulty is to use the fact that the original quantity of lead was in radioactive equilibrium with the larger amount of radium in the ore from which the metal was extracted. Let us, therefore, take samples of different ores and count the number of disintegrations of the radium in the ores. This was done for a variety of ores and the results are given in Table 1 below. These numbers vary from 0. Consequently, the number of disintegrations of the lead per minute per gram of white lead at the time of manufacture will vary from 0.
This implies that yo will also vary over a very large interval, since the number of disintegrations of lead is proportional to the amount present.
Thus, we cannot use Equation 5 to obtain an accurate, or even a crude estimate, of the age of a painting. However, we can still use Equation 5 to distinguish between a 17th century painting and a modern forgery. The basis for this statement is the simple observation that if the paint is very old compared to the 22 year half-life of lead, then the amount of radioactivity from the lead in the paint will be nearly equal to the amount of radioactivity from the radium in the paint.
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Differential Equations and Their Applications
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Differential equations and their applications solutions manual
Front Matter. Pages N2-xiv. PDF · First Order Differential Equations. Martin Braun. Pages PDF · Second Order Linear Differential Equations. Martin Braun.
Solution Manual book. Write a review. Our interactive player makes it easy to find solutions to An Introduction to Differential Equations and Their Applications problems you're working on - just go to the chapter for your book. Second order di erential equations reducible to rst order di erential equations 42 Chapter 4. General theory of di erential equations of rst order 45 Slope elds or direction elds 45 Autonomous rst order di erential equations. Dennis G.