Solutions manual advanced engineering
mathematics 10th edition by erwin
kreyszig
,PREFACE
General Character and Purpose of the Instructor's Manual
This Manual contains:
(I) Detailed solutions of the even-numbered problems.
(II) General comments on the purpose of each section and its classroom use, with math-
ematical and didactic information on teaching practice and pedagogical aspects. Some of
the comments refer to whole chapters (and are indicated accordingly).
Changes in Problem Sets
The major changes in this edition of the text are listed and explained in the Preface of the
book. They include global improvements by updating and streamlining chapters as well
as many local improvements aimed at simplification of the whole text. Speedy orienta-
tion is helped by chapter summaries at the end of the chapters, as in the last edition, and
by subdividing sections into subsections with unnumbered headings. Resulting effects of
these changes on the problem sets are as follows.
The problems have been changed. The large total number of over 4000 problems has
been retained, increasing their overall usefulness by. the following.
(I Balancing by extending problem sets that seemed too short and contracting others
that were too long, adjusting the length to the relative importance of the material in
a section, so that important issues are reflected sufficiently well not only in the text but
also in the problems. Thus, the danger of overemphasizing minor techniques and ideas is
avoided as much as possible.
(II) Simplification by omitting a small number of very difficult problems that appeared
in the previous edition, retaining the wide spectrum ranging from simple routine prob-
lems to more sophisticated engineering applications, and taking into account the "algo-
rithmic thinking" that is developing along with computers.
(ill) Close amalgamation of text, examples, and problems. This has again been
achieved by the large number of over 600 worked-out examples in the text and by in-
cluding problems closely related to those examples.
(IV) Addition of TEAM PROJECTS, CAS PROJECTS, and WRITING PROJ-
ECTS, whose role is explained in the Preface of the book under Big Changes.
These changes in the problem sets will help students in solving problems as well as in
gaining a better understanding of practical aspects in the text. It will also enable instruc-
tors to explain ideas and methods in terms of examples supplementing and illustrating
theoretical discussions-or even replacing some of them if so desired.
"Show the details of your work."
This request repeatedly stated in the book applies to all the problem sets. Of course, it is
intended to prevent the student from simply producing answers by a CAS instead of try-
ing to understand the underlying mathematics.
Orientation on Computers
Comments on computer use are included in the Preface of the book. Software systems are
listed in the book subsequent to Contents and at the beginning of Chap. 17 on numerical
methods.
ERWIN KRE
,IFFERENTIAL EQUATIONS
CHAPTER 1 First-Order Differential Equations
Major Changes
Direction fields are now discussed much earlier, in Sec. 1.2. This "geometrical" and "qual-
itative" approach to differential equations may provide a better conceptual under standing
of equations and solutions. The graphical power of a CAS will be helpful in this context.
The second major change concerns the combination of related solution methods. Solu-
tion by separation and solution by reduction to separable form now appear in a single sec-
tion (Sec. 1.3). Similarly, exact equations and integrating factors are both discussed in the
same section (Sec. 1.5).
Team Projects and CAS Projects are included in most problem sets.
SECTION 1.1. Basic Concepts and Ideas, page 2
Purpose. To give the student a first impression of what a differential equation is and
what we mean by solving it.
Background Material. For the whole chapter we need integration formulas and tech-
niques, which the student should review.
General Comments
This section should be covered relatively rapidly to get quickly to the actual solution meth-
ods in the next sections.
If an example of a partial differential equation is wanted in passing, Laplace's equa-
tion
a a
4 _z(]
a+? y
may be best because of its great physical importance.
Problem Set 1.1 is supposed to help the student with the tasks of
Solving y' = f(x) by calculus,
Finding particular solutions from given general solutions,
Setting up a differential equation for a given function as solution,
Gaining a first experience in modeling, by doing one or two problems,
Gaining a first impression of the importance of differential equations,
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 all solutions, that is, both the particular and the sin-
gular ones. We do not adopt this definition for two reasons. First, it is frequently quite
1
, i
2 Instructor's Manual
difficult to prove that a formula includes all solutions, 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 (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.
Comment on Example 2
Theoretically inclined students may show (a) by differentiation, (b) directly from the
differential equation, that the solution cannot be continued to the closed interval
-1 =x= 1, where the function is still continuous, but no longer differentiable.
This also illustrates that open intervals generally are the appropriate domains of defin-
ition of solutions.
SOLUTIONS TO PROBLEM SET 1.1, page 8
2. -(cos 3x)/3 + c 4. -le"
+e
10.x -yy'= 0 by implicit differentiation and division by 2.
12. From the solution and the initial condition, 0 + 1 = c. Answer: + y4 = 1 (y > 0).
The figure shows the portion of this curve in the first quadrant, together with a
quarter-circle for comparison.
OO 0.2 0.4 0.6 0.8 1 3x
Section 1.1. Problem 12
14.y= -r/4 16.y= (/2) secx
18. We get an ellipse with semi-axes la] and b = [al/2; that is, rla? + y(a/2)? =
1.
20. e-36 = 1/2, k = 0.192 541,e = 0.825 after 1 day, 3.012 · 10 after 365 days.
22.y"=g.By two integrations y' = gt + c, with c, = 0 because the stone starts from
rest, s =y= gr/2 + c with cs = 0 because s(0) = 0, the stone starts at s = 0.
24. k follows from %18oook 1/2, k = In (1/2)/18000 = -0.000038 508. Answer:
~35ooo 0.26y%. Since the decay is exponential, 36000 = 2· 18000 would give
(%/2)/2 = 0.25y%.
26.y' = ry, where r = 0.08; y(l) equals
1080.00, 1082.43, 1083.28, 1083.29
and y(5) equals
1469.33, 1485.95, 1491.76, 1491.82.
The last two numbers in each line differ only slightly from each other, as claimed.
mathematics 10th edition by erwin
kreyszig
,PREFACE
General Character and Purpose of the Instructor's Manual
This Manual contains:
(I) Detailed solutions of the even-numbered problems.
(II) General comments on the purpose of each section and its classroom use, with math-
ematical and didactic information on teaching practice and pedagogical aspects. Some of
the comments refer to whole chapters (and are indicated accordingly).
Changes in Problem Sets
The major changes in this edition of the text are listed and explained in the Preface of the
book. They include global improvements by updating and streamlining chapters as well
as many local improvements aimed at simplification of the whole text. Speedy orienta-
tion is helped by chapter summaries at the end of the chapters, as in the last edition, and
by subdividing sections into subsections with unnumbered headings. Resulting effects of
these changes on the problem sets are as follows.
The problems have been changed. The large total number of over 4000 problems has
been retained, increasing their overall usefulness by. the following.
(I Balancing by extending problem sets that seemed too short and contracting others
that were too long, adjusting the length to the relative importance of the material in
a section, so that important issues are reflected sufficiently well not only in the text but
also in the problems. Thus, the danger of overemphasizing minor techniques and ideas is
avoided as much as possible.
(II) Simplification by omitting a small number of very difficult problems that appeared
in the previous edition, retaining the wide spectrum ranging from simple routine prob-
lems to more sophisticated engineering applications, and taking into account the "algo-
rithmic thinking" that is developing along with computers.
(ill) Close amalgamation of text, examples, and problems. This has again been
achieved by the large number of over 600 worked-out examples in the text and by in-
cluding problems closely related to those examples.
(IV) Addition of TEAM PROJECTS, CAS PROJECTS, and WRITING PROJ-
ECTS, whose role is explained in the Preface of the book under Big Changes.
These changes in the problem sets will help students in solving problems as well as in
gaining a better understanding of practical aspects in the text. It will also enable instruc-
tors to explain ideas and methods in terms of examples supplementing and illustrating
theoretical discussions-or even replacing some of them if so desired.
"Show the details of your work."
This request repeatedly stated in the book applies to all the problem sets. Of course, it is
intended to prevent the student from simply producing answers by a CAS instead of try-
ing to understand the underlying mathematics.
Orientation on Computers
Comments on computer use are included in the Preface of the book. Software systems are
listed in the book subsequent to Contents and at the beginning of Chap. 17 on numerical
methods.
ERWIN KRE
,IFFERENTIAL EQUATIONS
CHAPTER 1 First-Order Differential Equations
Major Changes
Direction fields are now discussed much earlier, in Sec. 1.2. This "geometrical" and "qual-
itative" approach to differential equations may provide a better conceptual under standing
of equations and solutions. The graphical power of a CAS will be helpful in this context.
The second major change concerns the combination of related solution methods. Solu-
tion by separation and solution by reduction to separable form now appear in a single sec-
tion (Sec. 1.3). Similarly, exact equations and integrating factors are both discussed in the
same section (Sec. 1.5).
Team Projects and CAS Projects are included in most problem sets.
SECTION 1.1. Basic Concepts and Ideas, page 2
Purpose. To give the student a first impression of what a differential equation is and
what we mean by solving it.
Background Material. For the whole chapter we need integration formulas and tech-
niques, which the student should review.
General Comments
This section should be covered relatively rapidly to get quickly to the actual solution meth-
ods in the next sections.
If an example of a partial differential equation is wanted in passing, Laplace's equa-
tion
a a
4 _z(]
a+? y
may be best because of its great physical importance.
Problem Set 1.1 is supposed to help the student with the tasks of
Solving y' = f(x) by calculus,
Finding particular solutions from given general solutions,
Setting up a differential equation for a given function as solution,
Gaining a first experience in modeling, by doing one or two problems,
Gaining a first impression of the importance of differential equations,
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 all solutions, that is, both the particular and the sin-
gular ones. We do not adopt this definition for two reasons. First, it is frequently quite
1
, i
2 Instructor's Manual
difficult to prove that a formula includes all solutions, 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 (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.
Comment on Example 2
Theoretically inclined students may show (a) by differentiation, (b) directly from the
differential equation, that the solution cannot be continued to the closed interval
-1 =x= 1, where the function is still continuous, but no longer differentiable.
This also illustrates that open intervals generally are the appropriate domains of defin-
ition of solutions.
SOLUTIONS TO PROBLEM SET 1.1, page 8
2. -(cos 3x)/3 + c 4. -le"
+e
10.x -yy'= 0 by implicit differentiation and division by 2.
12. From the solution and the initial condition, 0 + 1 = c. Answer: + y4 = 1 (y > 0).
The figure shows the portion of this curve in the first quadrant, together with a
quarter-circle for comparison.
OO 0.2 0.4 0.6 0.8 1 3x
Section 1.1. Problem 12
14.y= -r/4 16.y= (/2) secx
18. We get an ellipse with semi-axes la] and b = [al/2; that is, rla? + y(a/2)? =
1.
20. e-36 = 1/2, k = 0.192 541,e = 0.825 after 1 day, 3.012 · 10 after 365 days.
22.y"=g.By two integrations y' = gt + c, with c, = 0 because the stone starts from
rest, s =y= gr/2 + c with cs = 0 because s(0) = 0, the stone starts at s = 0.
24. k follows from %18oook 1/2, k = In (1/2)/18000 = -0.000038 508. Answer:
~35ooo 0.26y%. Since the decay is exponential, 36000 = 2· 18000 would give
(%/2)/2 = 0.25y%.
26.y' = ry, where r = 0.08; y(l) equals
1080.00, 1082.43, 1083.28, 1083.29
and y(5) equals
1469.33, 1485.95, 1491.76, 1491.82.
The last two numbers in each line differ only slightly from each other, as claimed.
