SOLUTIONS MANUAL
FOR
ADVANCED ENGINEERING MATHEMATICS
TENTH EDITION
ERWIN KREYSZIG
HERBERT KREYSZI G
EDWARD J. NORMINTO N 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 initialvalue 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 furtherapplications in later sections)
Example 4: the straightforward solution of an IVPExample 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-orderlinear 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 laterwhen we actually need it); it is not too difficult to verify that a solution is
with arbitrary constants a,b,c,d.y/H11005ax/H11001b
cx/H11001d
1im01_demo.qxd 8/2/10 7:57 PM Page 1 Problem Set 1.1 will help the student with the tasks of
Solving by calculus
Finding particular solutions from given general solutionsSetting up an ODE for a given function as solution, e.g., Gaining a first experience in modeling, by doing one or two problemsGaining 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 frequentlyquite difficult to prove that a formula includes allsolutions; hence, this definition of a
general solution is rather useless in practice. Second, linear differential equations
(satisfying rather general conditions on the coefficients) have no singular solutions (asmentioned in the text), so that for these equations a general solution as defined does includeall solutions. For the latter reason, some books use the term “general solution” for linearequations 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)>2/H110050.25y036,000 /H110052#18,000 e35,000 k/H110050.26y0e18,000 k/H110051
2, k/H11005(ln 1
2)>18,000 /H11005/H11002 0.000039e/H115463.6k/H110051
2˛, k/H110050.19254, (a) e/H11546k/H110050.825, (b) 3.012#10/H1154631y/H110051
4 x2, yr/H110051
2 x, thus 1
4 x2/H11002x (1
2 x)/H110011
4 x2/H110050yr2/H11002xyr/H11001y/H11005c2/H11002xc/H11001(cx/H11002c2) /H110050y/H11005cx/H11002c2y/H110054/H110024 sin2 xy2/H110024x2/H1100512y/H11005pe/H115462.5x2y/H11005/H110021
0.23 e/H115460.2x/H11001c1x2/H11001c2x/H11001c3y/H11005a cos x/H11001b sin xy/H11005ce/H115461.5xy/H11005e/H11546x2>2/H11001cy/H11005exyr/H11005f (x)2 Instructor’s Manualim01_demo.qxd 8/2/10 7:57 PM Page 2 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 formalmanipulations of solution methods, the latter being restricted to relatively small—albeitimportant—classes of ODEs. This approach is becoming increasingly important, especiallybecause 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 differentiationand 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 geometricunderstanding 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 1.2 will help the student with the tasks of:
Drawing direction fields and approximate solution curves
Handling your CAS in selecting appropriate windows for specific tasksA first look at the important Verhulst equation (Prob. 4)Bell-shaped curves as solutions of a simple ODEOutflow 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.
12. , not needed to do the problem.
14. , not needed to do the problem.
16. (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 usuallycritical and often tend to give nonsense.ƒ yrƒy(x) /H11005sin (x/H110011
4 p)y/H1100521t/H110011y/H11005ce/H11546x2y(x)/H11005/H11002 arctan [1>(x/H11001c)](0, 3) (0, 1)[(0, 0) and (0, 2)]x2/H110011
4 y2/H11005cyr/H11549f (x, y)Instructor’s Manual 3im01_demo.qxd 8/2/10 7:57 PM Page 3
FOR
ADVANCED ENGINEERING MATHEMATICS
TENTH EDITION
ERWIN KREYSZIG
HERBERT KREYSZI G
EDWARD J. NORMINTO N 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 initialvalue 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 furtherapplications in later sections)
Example 4: the straightforward solution of an IVPExample 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-orderlinear 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 laterwhen we actually need it); it is not too difficult to verify that a solution is
with arbitrary constants a,b,c,d.y/H11005ax/H11001b
cx/H11001d
1im01_demo.qxd 8/2/10 7:57 PM Page 1 Problem Set 1.1 will help the student with the tasks of
Solving by calculus
Finding particular solutions from given general solutionsSetting up an ODE for a given function as solution, e.g., Gaining a first experience in modeling, by doing one or two problemsGaining 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 frequentlyquite difficult to prove that a formula includes allsolutions; hence, this definition of a
general solution is rather useless in practice. Second, linear differential equations
(satisfying rather general conditions on the coefficients) have no singular solutions (asmentioned in the text), so that for these equations a general solution as defined does includeall solutions. For the latter reason, some books use the term “general solution” for linearequations 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)>2/H110050.25y036,000 /H110052#18,000 e35,000 k/H110050.26y0e18,000 k/H110051
2, k/H11005(ln 1
2)>18,000 /H11005/H11002 0.000039e/H115463.6k/H110051
2˛, k/H110050.19254, (a) e/H11546k/H110050.825, (b) 3.012#10/H1154631y/H110051
4 x2, yr/H110051
2 x, thus 1
4 x2/H11002x (1
2 x)/H110011
4 x2/H110050yr2/H11002xyr/H11001y/H11005c2/H11002xc/H11001(cx/H11002c2) /H110050y/H11005cx/H11002c2y/H110054/H110024 sin2 xy2/H110024x2/H1100512y/H11005pe/H115462.5x2y/H11005/H110021
0.23 e/H115460.2x/H11001c1x2/H11001c2x/H11001c3y/H11005a cos x/H11001b sin xy/H11005ce/H115461.5xy/H11005e/H11546x2>2/H11001cy/H11005exyr/H11005f (x)2 Instructor’s Manualim01_demo.qxd 8/2/10 7:57 PM Page 2 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 formalmanipulations of solution methods, the latter being restricted to relatively small—albeitimportant—classes of ODEs. This approach is becoming increasingly important, especiallybecause 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 differentiationand 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 geometricunderstanding 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 1.2 will help the student with the tasks of:
Drawing direction fields and approximate solution curves
Handling your CAS in selecting appropriate windows for specific tasksA first look at the important Verhulst equation (Prob. 4)Bell-shaped curves as solutions of a simple ODEOutflow 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.
12. , not needed to do the problem.
14. , not needed to do the problem.
16. (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 usuallycritical and often tend to give nonsense.ƒ yrƒy(x) /H11005sin (x/H110011
4 p)y/H1100521t/H110011y/H11005ce/H11546x2y(x)/H11005/H11002 arctan [1>(x/H11001c)](0, 3) (0, 1)[(0, 0) and (0, 2)]x2/H110011
4 y2/H11005cyr/H11549f (x, y)Instructor’s Manual 3im01_demo.qxd 8/2/10 7:57 PM Page 3
