Solucionario Ecuaciones Diferenciales

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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 solutionCa>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
valuesgreater 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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solution Ca>b œ ! . For C  ! , the slopes are all positive; thus solutions with initial
valuesless than ! approach the solution Ca>b œ ! .
14.

Observe 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 greater than # diverge from Ca>b œ # . For !  C  #, the slopes are also
positive, and hence solutions with initial values between ! and # all increasetoward 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 Q a>b be the total amount of the drug ain milligramsb in the patient's body at
any
given time > a2

a71Î2b œ >  $ is a solution.
20.

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All solutions approach the equilibrium solution Ca>b œ ! Þ
23.

All solutions appear to diverge from the sinusoid Ca>b œ 

$
È#

=38Ð>  1 Ñ  " ,
%

which is also a solution corresponding to the initial value Ca!b œ  &Î# .
25.

All solutions appear to converge to Ca>b œ ! . First, the rate of change is small. The
slopes
eventually increasevery rapidly in magnitude.
26.

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The direction field is rather complicated. Nevertheless, the collection of points at which
the slope field is zero, is given by the implicit equation C$  'C œ #># Þ The graph of
these points is shown below:

The y-intercepts...