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—————————————————————————— CHAPTER 1. ——
Chapter One
Section 1.1
1.
For C "Þ& , the slopes are negative, and hence the solutions decrease. For C "Þ& , the
slopes are positive, and hence the solutions increase. The equilibrium solution appears
to be Ca>b œ "Þ& , to which all other solutions converge.
3.
For C "Þ& , the slopes are :9=3tive, and hence the solutions increase. For C "Þ&
, the slopes are negative, and hence the solutions decrease. All solutions appear to
diverge awaỵ from the equilibrium solution Ca>b œ "Þ& .
5.
For C "Î# , the slopes are :9=3tive, and hence the solutions increase. For
C "Î# , the slopes are negative, and hence the solutions decrease. All solutions diverge
awaỵ from
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—————————————————————————— CHAPTER 1. ——
the equilibrium solution Ca>b œ "Î# .
6.
For C # , the slopes are :9=3tive, and hence the solutions increase. For C # ,
the slopes are negative, and hence the solutions decrease. All solutions diverge awaỵ
from
the equilibrium solution Ca>b œ # .
8. For all solutions to approach the equilibrium solution Ca>b œ #Î$ , we must have
C w ! for C #Î$ , and C w ! for C #Î$ . The required rates are satisfied bỵ the
differential equation C w œ # $C .
9. For solutions other than Ca>b œ # to diverge from C œ # , Ca>b must be an increasing
function for C # , and a decreasing function for C # . The simplest differential equation
whose solutions satisfỵ these criteria is C w œ C # .
10. For solutions other than Ca>b œ "Î$ to diverge from C œ "Î$ , we must have C w !
for C "Î$ , and C w ! for C "Î$ . The required rates are satisfied bỵ the differential
equation C w œ $C " .
12.
Note that C w œ ! for C œ ! and C œ & . The two equilibrium solutions are Ca>b œ ! and
Ca>b œ & . Based on the direction field, C w ! for C & ; thus solutions with initial
values greater than & diverge from the solution Ca>b œ & . For ! C &, the slopes are
negative, and hence solutions with initial values between ! and & all decrease toward
the
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—————————————————————————— CHAPTER 1. ——
solution Ca>b œ ! . For C ! , the slopes are all positive; thus solutions with initial
values
less than ! approach the solution Ca>b œ ! .
14.
Observe that C w œ ! for C œ ! and C œ # . The two equilibrium solutions are Ca>b œ !
and Ca>b œ # . Based on the direction field, C w ! for C # ; thus solutions with initial
values greater than # diverge from Ca>b œ # . For ! C #, the slopes are also
positive, and hence solutions with initial values between ! and # all increase toward the
solution
Ca>b œ # . For C ! , the slopes are all negative; thus solutions with initial
values less than ! diverge from the solution Ca>b œ ! .
16. a+b Let Qa>b be the total amount of the drug ain milligramsb in the patient's bodỵ at
anỵ
given time > a2<=b . The drug is administered into the bodỵ at a constant rate of &!!
71Î2<Þ
The rate at which the drug leaves the bloodstream is given bỵ !Þ%Qa>b Þ Hence the
accumulation rate of the drug is described bỵ the differential equation
.Q
œ &!! !Þ% Q a71Î2<b Þ
.>
a,b
Based on the direction field, the amount of drug in the bloodstream approaches the
equilibrium level of "#&! 71 aA3>238 + 0 /A 29?<=bÞ
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—————————————————————————— CHAPTER 1. ——
Chapter One
Section 1.1
1.
For C "Þ& , the slopes are negative, and hence the solutions decrease. For C "Þ& , the
slopes are positive, and hence the solutions increase. The equilibrium solution appears
to be Ca>b œ "Þ& , to which all other solutions converge.
3.
For C "Þ& , the slopes are :9=3tive, and hence the solutions increase. For C "Þ&
, the slopes are negative, and hence the solutions decrease. All solutions appear to
diverge awaỵ from the equilibrium solution Ca>b œ "Þ& .
5.
For C "Î# , the slopes are :9=3tive, and hence the solutions increase. For
C "Î# , the slopes are negative, and hence the solutions decrease. All solutions diverge
awaỵ from
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1
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—————————————————————————— CHAPTER 1. ——
the equilibrium solution Ca>b œ "Î# .
6.
For C # , the slopes are :9=3tive, and hence the solutions increase. For C # ,
the slopes are negative, and hence the solutions decrease. All solutions diverge awaỵ
from
the equilibrium solution Ca>b œ # .
8. For all solutions to approach the equilibrium solution Ca>b œ #Î$ , we must have
C w ! for C #Î$ , and C w ! for C #Î$ . The required rates are satisfied bỵ the
differential equation C w œ # $C .
9. For solutions other than Ca>b œ # to diverge from C œ # , Ca>b must be an increasing
function for C # , and a decreasing function for C # . The simplest differential equation
whose solutions satisfỵ these criteria is C w œ C # .
10. For solutions other than Ca>b œ "Î$ to diverge from C œ "Î$ , we must have C w !
for C "Î$ , and C w ! for C "Î$ . The required rates are satisfied bỵ the differential
equation C w œ $C " .
12.
Note that C w œ ! for C œ ! and C œ & . The two equilibrium solutions are Ca>b œ ! and
Ca>b œ & . Based on the direction field, C w ! for C & ; thus solutions with initial
values greater than & diverge from the solution Ca>b œ & . For ! C &, the slopes are
negative, and hence solutions with initial values between ! and & all decrease toward
the
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—————————————————————————— CHAPTER 1. ——
solution Ca>b œ ! . For C ! , the slopes are all positive; thus solutions with initial
values
less than ! approach the solution Ca>b œ ! .
14.
Observe that C w œ ! for C œ ! and C œ # . The two equilibrium solutions are Ca>b œ !
and Ca>b œ # . Based on the direction field, C w ! for C # ; thus solutions with initial
values greater than # diverge from Ca>b œ # . For ! C #, the slopes are also
positive, and hence solutions with initial values between ! and # all increase toward the
solution
Ca>b œ # . For C ! , the slopes are all negative; thus solutions with initial
values less than ! diverge from the solution Ca>b œ ! .
16. a+b Let Qa>b be the total amount of the drug ain milligramsb in the patient's bodỵ at
anỵ
given time > a2<=b . The drug is administered into the bodỵ at a constant rate of &!!
71Î2<Þ
The rate at which the drug leaves the bloodstream is given bỵ !Þ%Qa>b Þ Hence the
accumulation rate of the drug is described bỵ the differential equation
.Q
œ &!! !Þ% Q a71Î2<b Þ
.>
a,b
Based on the direction field, the amount of drug in the bloodstream approaches the
equilibrium level of "#&! 71 aA3>238 + 0 /A 29?<=bÞ
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3
