Resolução boyce cap. i

—————————————————————————— 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 away 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 away 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 away 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 by the differential equation C w œ #  $C .

9. For solutions other than Ca>b œ # to diverge from C œ # , C a>b must be an increasing function for C  # , and a decreasing function for C  # . The simplest differential equation whose solutions satisfy 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 by the differential equation C w œ $C  " .
12.

Note that C w œ ! for C œ ! and C œ & . The two equilibrium solutions are C a>b œ ! and
Ca>b œ & . Based on the direction field, C w  ! for C  & ; thus solutions with initial
values