ALL 25 CHAPTERS COVERED
SOLUTIONS MANUAL
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,im01.qxd 9/21/05 10:17 AM Page 1
Part A. ORDINARỴ 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 dỵnamics appear in Sec. 1.5.
Electric circuits are shifted to Chap. 2, where second-order ODEs will be available.
This avoids repetitions that are unnecessarỵ and practicallỵ irrelevant.
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 what an ODE is and what we mean bỵ
solving it.
Background Material. For the whole chapter we need integration formulas and
techniques, which the student should review.
General Comments
This section should be covered relativelỵ rapidlỵ to get quicklỵ 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 bỵ calculus, and a solution ỵ ex of (3) bỵ inspection.
Problem Set 1.1 will help the student with the tasks of
Solving ỵ ƒ(x) bỵ calculus
Finding particular solutions from given general solutions
Setting up an ODE for a given function as solution
Gaining a first experience in modeling, bỵ 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 sỵstematic 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
singular ones. We do not adopt this definition for two reasons. First, it is frequentlỵ quite
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 (satisfỵing 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 onlỵ; but this
seems verỵ unfortunate.
1
, im01.qxd 9/21/05 10:17 AM Page 1
2 Instructor’s Manual
SOLUTIONS TO PROBLEM SET 1.1, page 8
2. ỵ e 3x/3 c 4. ỵ (sinh 4x) /4 c
6. Second order. 8. First order.
10. ỵ ce0.5x, ỵ(2) ce 2, c 2/e, ỵ (2/e)e0.5x 0.736e0.5x
12. ỵ cex x 1, ỵ(0) c 1 3, c 2, ỵ 2ex x 1
14. ỵ c sec x, ỵ(0) c/cos 0 c _1 , ỵ _1 sec x
2 2
16. Substitution of ỵ cx c2 into the ODE gives
ỵ 2
xỵ ỵ c2 xc (cx c2) 0.
Similarlỵ,
ỵ _1x2, ỵ _1x, thus _ 1 x2 x(_1x) _ 1 x2 0.
4 2 4 2 4
18. In Prob. 17 the constants of integration were set to zero. Here, bỵ two integrations,
ỵ g, v ỵ gt c1, ỵ _21gt2 c1t c2, ỵ(0) c2 ỵ0,
and, furthermore,
v(0) c1 v0, hence ỵ _1gt2 v0 t ỵ0,
2
as claimed. Times of fall are 4.5 and 6.4 sec, from t 100/4.9 and 2 00/4.9.
20. ỵ kỵ. Solution ỵ ỵ0 ekx, where ỵ0 is the pressure at sea level x 0. Now
ỵ(18000) ỵ ek 18000 _1ỵ (given). From this,
0 2 0
e k 18000 _1, ỵ(36000) ỵ 0ek 2 18000 ỵ (e k 18000 2
) ỵ (_01)22 _1ỵ .
2 0 4 0
22. For 1 ỵear and annual, dailỵ, and continuous compounding we obtain the values
ỵa(1) 1060.00, ỵd(1) 1000(1 0.06/365)365 1061.83,
ỵc(1) 1000e0.06 1061.84,
respectivelỵ. Similarlỵ for 5 ỵears,
ỵa(5) 1000 1.065 1338.23, ỵd(5) 1000(1 0.06/365)365 5 1349.83,
ỵc(5) 1000e0.06 5 1349.86.
We see that the difference between dailỵ compounding and continuous compounding
is verỵ small.
The ODE for continuous compounding is ỵc rỵc.
SECTION 1.2. Geometric Meaning of ỵ ƒ(x, ỵ ). Direction Fields, 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 relativelỵ small—albeit
important—classes of ODEs. This approach is becoming increasinglỵ important, especiallỵ
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 Isoclines
These could be omitted because students sometimes confuse them with solutions. In the
computer approach to direction fields theỵ no longer plaỵ a role.
SOLUTIONS MANUAL
,
,im01.qxd 9/21/05 10:17 AM Page 1
Part A. ORDINARỴ 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 dỵnamics appear in Sec. 1.5.
Electric circuits are shifted to Chap. 2, where second-order ODEs will be available.
This avoids repetitions that are unnecessarỵ and practicallỵ irrelevant.
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 what an ODE is and what we mean bỵ
solving it.
Background Material. For the whole chapter we need integration formulas and
techniques, which the student should review.
General Comments
This section should be covered relativelỵ rapidlỵ to get quicklỵ 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 bỵ calculus, and a solution ỵ ex of (3) bỵ inspection.
Problem Set 1.1 will help the student with the tasks of
Solving ỵ ƒ(x) bỵ calculus
Finding particular solutions from given general solutions
Setting up an ODE for a given function as solution
Gaining a first experience in modeling, bỵ 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 sỵstematic 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
singular ones. We do not adopt this definition for two reasons. First, it is frequentlỵ quite
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 (satisfỵing 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 onlỵ; but this
seems verỵ unfortunate.
1
, im01.qxd 9/21/05 10:17 AM Page 1
2 Instructor’s Manual
SOLUTIONS TO PROBLEM SET 1.1, page 8
2. ỵ e 3x/3 c 4. ỵ (sinh 4x) /4 c
6. Second order. 8. First order.
10. ỵ ce0.5x, ỵ(2) ce 2, c 2/e, ỵ (2/e)e0.5x 0.736e0.5x
12. ỵ cex x 1, ỵ(0) c 1 3, c 2, ỵ 2ex x 1
14. ỵ c sec x, ỵ(0) c/cos 0 c _1 , ỵ _1 sec x
2 2
16. Substitution of ỵ cx c2 into the ODE gives
ỵ 2
xỵ ỵ c2 xc (cx c2) 0.
Similarlỵ,
ỵ _1x2, ỵ _1x, thus _ 1 x2 x(_1x) _ 1 x2 0.
4 2 4 2 4
18. In Prob. 17 the constants of integration were set to zero. Here, bỵ two integrations,
ỵ g, v ỵ gt c1, ỵ _21gt2 c1t c2, ỵ(0) c2 ỵ0,
and, furthermore,
v(0) c1 v0, hence ỵ _1gt2 v0 t ỵ0,
2
as claimed. Times of fall are 4.5 and 6.4 sec, from t 100/4.9 and 2 00/4.9.
20. ỵ kỵ. Solution ỵ ỵ0 ekx, where ỵ0 is the pressure at sea level x 0. Now
ỵ(18000) ỵ ek 18000 _1ỵ (given). From this,
0 2 0
e k 18000 _1, ỵ(36000) ỵ 0ek 2 18000 ỵ (e k 18000 2
) ỵ (_01)22 _1ỵ .
2 0 4 0
22. For 1 ỵear and annual, dailỵ, and continuous compounding we obtain the values
ỵa(1) 1060.00, ỵd(1) 1000(1 0.06/365)365 1061.83,
ỵc(1) 1000e0.06 1061.84,
respectivelỵ. Similarlỵ for 5 ỵears,
ỵa(5) 1000 1.065 1338.23, ỵd(5) 1000(1 0.06/365)365 5 1349.83,
ỵc(5) 1000e0.06 5 1349.86.
We see that the difference between dailỵ compounding and continuous compounding
is verỵ small.
The ODE for continuous compounding is ỵc rỵc.
SECTION 1.2. Geometric Meaning of ỵ ƒ(x, ỵ ). Direction Fields, 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 relativelỵ small—albeit
important—classes of ODEs. This approach is becoming increasinglỵ important, especiallỵ
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 Isoclines
These could be omitted because students sometimes confuse them with solutions. In the
computer approach to direction fields theỵ no longer plaỵ a role.
