EDWARD J. NORMINTO - EQUATIONS (ODEs) CHAPTER 1 First-Order ODEs Maj...

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OLUTIONS MANUAL

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ENGINEERING

MATHEMATICS

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ERWIN KREYSZIG

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ERBERT KREYSZIG

EDWARD J. NORMINTO

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Part A. ORDINARY DIFFERENTIAL EQUATIONS (ODEs) CHAPTER 1 First-Order ODEs Major Changes There is more material on modeling in the text as well as in the problem set.Some additions on population dynamics appear in Sec. 1.5.Team Projects, CAS Projects, and CAS Experiments are included in most problem sets.SECTION 1.1. Basic Concepts. Modeling, page 2 Purpose.To give the students a first impression of what an ODE is and what we mean by solving it.The role of initial conditions should be emphasized since, in most cases, solving an engineering problem of a physical nature usually means finding the solution of an initial value problem (IVP).

Further points to stress and illustrate by examples are:

The fact that a general solution represents a family of curves.The distinction between an arbitrary constant, which in this chapter will always be denoted by c, and a fixed constant (usually of a physical or geometric nature and given in most cases).The examples of the text illustrate the following.

Example 1: the verification of a solution

Examples 2 and 3: ODEs that can actually be solved by calculus with Example 2 giving an impression of exponential growth (Malthus!) and decay (radioactivity and further applications in later sections)

Example 4: the straightforward solution of an IVP

Example 5: a very detailed solution in all steps of a physical IVP involving a physical constant k Background Material.For the whole chapter we need integration formulas and techniques from calculus, which the student should review.General Comments on Text This section should be covered relatively rapidly to get quickly to the actual solution methods in the next sections.Equations (1)–(3) are just examples, not for solution, but the student will see that solutions of (1) and (2) can be found by calculus. Instead of (3), one could perhaps take a third-order linear ODE with constant coefficients or an Euler–Cauchy equation, both not of great interest.The present (3) is included to have a nonlinear ODE (a concept that will be mentioned later when we actually need it); it is not too difficult to verify that a solution is with arbitrary constants a,b,c,d.y axb cxd 1

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Problem Set 1.1will help the student with the tasks of Solving by calculus Finding particular solutions from given general solutions Setting up an ODE for a given function as solution, e.g., Gaining a first experience in modeling, by doing one or two problems Gaining a first impression of the importance of ODEs without wasting time on matters that can be done much faster, once systematic methods are available.Comment on “General Solution” and “Singular Solution” Usage of the term “general solution” is not uniform in the literature. Some books use the term to mean a solution that includes allsolutions, that is, both the particular and the singular ones. We do not adopt this definition for two reasons. First, it is frequently quite difficult to prove that a formula includes allsolutions; hence, this definition of a general solution is rather useless in practice. Second, lineardifferential equations (satisfying rather general conditions on the coefficients) have no singular solutions (as mentioned in the text), so that for these equations a general solution as defined does include all solutions. For the latter reason, some books use the term “general solution” for linear equations only; but this seems very unfortunate.SOLUTIONS TO PROBLEM SET 1.1, page 8 2.

4.

6.

8.

10.

12.

14.

16.Substitution of into the ODE gives .Similarly, .

18..

20.kfollows from .

Answer:. Since the decay is exponential, would

give .(y 0>2)>20.25y

36,0002 #

18,000e 35,000k 0.26y

e 18,000k

1 2, k(ln 1 2)>18,0000.000039 e 3.6k

1 2 ˛, k0.19254, (a) e k 0.825, (b) 3.012# 10 31 y 1

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SECTION 1.2. Geometric Meaning of . Direction Fields, Euler’s Method, page 9 Purpose.To give the student a feel for the nature of ODEs and the general behavior of fields of solutions. This amounts to a conceptual clarification before entering into formal manipulations of solution methods, the latter being restricted to relatively small—albeit important—classes of ODEs. This approach is becoming increasingly important, especially because of the graphical power of computer software. It is the analog of conceptual studies of the derivative and integral in calculus as opposed to formal techniques of differentiation and integration.Comment on Order of Sections This section could equally well be presented later in Chap. 1, perhaps after one or two formal methods of solution have been studied.Euler’s method has been included for essentially two reasons, namely, as an early eye opener to the possibility of numerically obtaining approximate values of solutions by step-by-step computations and, secondly, to enhance the student’s conceptual geometric understanding of the nature of an ODE.Furthermore, the inaccuracy of the method will motivate the development of much more accurate methods by practically the same basic principle (in Sec. 21.1).

Problem Set 1.2will help the student with the tasks of:

Drawing direction fields and approximate solution curves Handling your CAS in selecting appropriate windows for specific tasks A first look at the important Verhulst equation (Prob. 4) Bell-shaped curves as solutions of a simple ODE Outflow from a vessel (analytically discussed in the next section) Discussing a few types of motion for given velocity (Parachutist, etc.) Comparing approximate solutions for different step size SOLUTIONS TO PROBLEM SET 1.2, page 11 2.Ellipses . If your CAS does not give you what you expected, change the given point.

4.Verhulst equation, to be discussed as a population model in Sec. 1.5. The given points correspond to constant solutions , an increasing solution through , and a decreasing solution through .

6.Solution , not needed for doing the problem.

8.ODE of the bell-shaped curves .

10.ODE of the outflow from a vessel, to be discussed in Sec. 1.3.

• , not needed to do the problem.

14., not needed to do the problem.

• (a)Your PC may give you fields of varying quality, depending on the choice of the

region graphed, and good choices are often obtained only after some trial and error.Enlarging generally gives more details. Subregions where is large are usually critical and often tend to give nonsense.ƒ yrƒ y(x) sin (x 1

• p)

y21t1 yce x 2 y(x)arctan [1>(xc)]

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[(0, 0) and (0, 2)] x 2

1

• y

2 c y rf (x, y) Instructor’s Manual3

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