James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

606 Chapter 9 Differential Equations
31. y − k x
32. y − 1
x 1 k
33–35 An integral equation is an equation that contains an
unknown function ysxd and an integral that involves ysxd. Solve the
given integral equation. [Hint: Use an initial condition obtained
from the integral equation.]
33. ysxd − 2 1 y x
ft 2 tystdg dt
34. ysxd − 2 1 y x
2
1
dt
tystd , x . 0
35. ysxd − 4 1 y x
2tsystd dt
0
36. Find a function f such that f s3d − 2 and
st 2 1 1df 9std 1 f f stdg 2 1 1 − 0 t ± 1
[Hint: Use the addition formula for tansx 1 yd on Reference
Page 2.]
37. Solve the initial-value problem in Exercise 9.2.27 to find an
expression for the charge at time t. Find the limiting value of
the charge.
38. In Exercise 9.2.28 we discussed a differential equation that
models the temperature of a 95°C cup of coffee in a 20°C
room. Solve the differential equation to find an expression for
the temperature of the coffee at time t.
39. In Exercise 9.1.15 we formulated a model for learning in the
form of the differential equation
dP
− ksM 2 Pd
dt
where Pstd measures the performance of someone learning a
skill after a training time t, M is the maximum level of performance,
and k is a positive constant. Solve this differential
equation to find an expression for Pstd. What is the limit of this
expression?
40. In an elementary chemical reaction, single molecules of
two reactants A and B form a molecule of the product C:
A 1 B l C. The law of mass action states that the rate of
reaction is proportional to the product of the concentrations of
A and B:
dfCg
− k fAgfBg
dt
(See Example 3.7.4.) Thus, if the initial concentrations are
fAg − a molesyL and fBg − b molesyL and we write x − fCg,
then we have
dx
− ksa 2 xdsb 2 xd
dt
(a) Assuming that a ± b, find x as a function of t. Use the fact
that the initial concentration of C is 0.
(b) Find xstd assuming that a − b. How does this expression
for xstd simplify if it is known that fCg − 1 2 a after
20 seconds?
41. In contrast to the situation of Exercise 40, experiments show
that the reaction H 2 1 Br 2 l 2HBr satisfies the rate law
dfHBrg
− k fH 2gfBr 2g 1y2
dt
and so for this reaction the differential equation becomes
dx
− ksa 2 xdsb 2 xd 1y2
dt
where x − fHBrg and a and b are the initial concentrations of
hydrogen and bromine.
(a) Find x as a function of t in the case where a − b. Use the
fact that xs0d − 0.
(b) If a . b, find t as a function of x. fHint: In performing the
integration, make the substitution u − sb 2 x.g
42. A sphere with radius 1 m has temperature 15 8C. It lies inside a
concentric sphere with radius 2 m and temperature 25 8C. The
temperature Tsrd at a distance r from the common center of
the spheres satisfies the differential equation
d 2 T
dr 2
1 2 r
dT
dr − 0
If we let S − dTydr, then S satisfies a first-order differential
equation. Solve it to find an expression for the temperature
Tsrd between the spheres.
43. A glucose solution is administered intravenously into the
bloodstream at a constant rate r. As the glucose is added, it
is converted into other substances and removed from the
bloodstream at a rate that is proportional to the concentration
at that time. Thus a model for the concentration C − Cstd of
the glucose solution in the bloodstream is
dC
− r 2 kC
dt
where k is a positive constant.
(a) Suppose that the concentration at time t − 0 is C 0. Determine
the concentration at any time t by solving the differential
equation.
(b) Assuming that C 0 , ryk, find lim t l ` Cstd and interpret
your answer.
44. A certain small country has $10 billion in paper currency in
circulation, and each day $50 million comes into the country’s
banks. The government decides to introduce new currency by
having the banks replace old bills with new ones whenever
old currency comes into the banks. Let x − xstd denote the
amount of new currency in circulation at time t, with xs0d − 0.
(a) Formulate a mathematical model in the form of an initialvalue
problem that represents the “flow” of the new currency
into circulation.
(b) Solve the initial-value problem found in part (a).
(c) How long will it take for the new bills to account for 90%
of the currency in circulation?
45. A tank contains 1000 L of brine with 15 kg of dissolved salt.
Pure water enters the tank at a rate of 10 Lymin. The solution
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