24, 2o t4
Jcr-nr,ra rq
U
432
APPENDIX
C
D
A
r
cotrlvts
Differential Equations
Ihe folloring warnrup exercises involve skills that were covered in earlier sectiqns.
You wilt use these s&ills in the exercise set fior this section-
ln kercises 1-6,find the indefinite integral and check your resuh by differentiating.
t
l- I *'t'*
J,
,
.,i
s.
(-tP +eDz:C
J
f
z. I (t' -
,t/t) dt
J
3.
fz
l--:
Jr-) _dx
"[r4*
f
s. I d'dy
J
f
6. I xet-" dx
J
ln Exercises 7-1Q solve the equation
7. (3F-
9.
br
C
or
&
6(3):1a6'
ta: D*
10. (6),
ln Exercises l-6, decide whether the variables in the differential
equation can be separated.
dY: *
,-'dx
y+3
+r
s.9:L
clx x
-dv
-:-:r-Y
5.
c8
fu x+y
a. r4:!
dxy
s.a:L
dxx
t.yz!=r
fi.4:
dx
tzx
11. (y
+
t)*: x
13.y'-ry=A
B.u:!
dt 4y
u.+:Jt-v
tlx
2y
20.
y'=
(2r
ry':,
x
23. v':x " y l+v
2s. d(y'+ 1) :
22.
y'-
y(r + t) =
19. (2 + x)y'
-
lxl, + 3)
26.
W'-
2xe*
:0
ln Exercises 27-32, use the initial condition to find the particular
solution of the differential equation-
Differential Equotion
Initial Condition
21.
W' - et:
2s.
J; + Jyy':o
y = 4when.r:0
/: 4whenx : I
O
29.x$t+4)+t':O
Y: -5 whenr:0
x2u
,o.*:
12.(r+f*-4x:o
31. dP
/:3whenr=0
P:5whenr:0
7: 140 when r :
14-
y'- y:
5
rc.
u,*:
3p
".*: ,8,
O
,n-*:#
1
ln Exercises 7-26, use separation of variables to find the general
solution of the differential equation.
clx
:
"-*
21.
t dy_x+l
dxx
t.d!: *
z1:zr
* 3(6) :
+t
xz(t + y)
-
6P
32. dT + k{T
dt: 0
- 70)dt:
o
0
APPENDIXC
ln Exercises 33 and 34,find an equation forthe graph that passes
through the point and has the specified slope.Then graph the
equation-
33. Point
Slope:
(-1, t)
6x
5y
y'
41.
air resistance.
Kto
-
N)
diferential equatioa.
12. Sales The rate of increase in sales J (in thousands of
units) ofa product is proportional to the current level of
sales and inversely proportional to the sguare of the time r.
This is described by tbe differential equation
ds rs
dtP
35.
e.s*:
43.2
-
1.25v
36.
D.s*:
43.2
-
t.75v
Chemistry: Newton's Law of Cooling
where t is the time in years. The saturation point for the
martet is 50,fiX) units. That is, the limit of ,S as r-+oo is
50. After I year, 10,(ffi ufts have been sold- Find S as a
firnction of tte time r.
ln Exercises 37-39,
tawof Cooling,which states that the rate of change
in the temperature f of an object is proportional to the difference
use NewtonS
between the temperature Fof the object and the temperature Io
of the surrounding environmentThis is described by the differential equation df /dt
IJ.
: (f -
37. A steel ingot whose temperature is 1500T is placed in a
room whose temperature is a constant 90iF. One hour later,
the temperature of the ingot is l l20"F. Wbat is the iogot's
temperature 5 hours after it is placed in the room?
38. A rooar is kept at a coastant temperature of 70"F. An object
placd ia the room cools &om 350.F to 150"f in 45 ainutes. How long will it take for the object to cool to a
t€xrryeratrre of 80"F?
39. Food at a tefirperature of 70T is placed in a freezer that is
set d 0T. After I hour, the tryrature of the food is 48T.
(a) Find the temperature ofthe food after it
has been in the
&eezer 6 hours-
Law
43. Economics: Pareto's
Ac*ordmg to tte economist\frlfredo Pareto (1848-1923), the rate ofdecrease of
the number of people y ia a stable economy having an
income of at least r dollars is directly pmportional to the
number ofsuch people and inversely proportional to their
income r. This is modeled by the differential equation
dv
-y
-::
dxx -k"-Solve this diflerential equation
44. Economics: Paruta's Law Ia 1998, 8.6millioupeople
in &e United States eamed more than $25,000 and 50.3
million people earned more than $25,000 (see figure).
Assume that Pareto's Lary holds and use the result of
Exercise 43 to determine the aumber of people (in millions) who earned (a) more than $20,000 and (b) more than
$100,000. (Source: U.S.
Census Bureau)
Pceto's
L.*
(b) How long will it take the food to cool to a tcmperahre
of l0T?
Bioleg:
Cell Grswth The rate of growth of a spherical cell with volume Zis pmportional to its surfrce alea S.
For a sphere, the surface area and volume are related by
S klr2/3. So, a model for the cell's growth is
:
#:
has
where I is &e time in days. Solve this
Yelacity ln Exercises 35 and 34 solve the differential equation
to find velocity y as a function of time t if y : 0 whe* f : 0. The
differential equation models the motion of two people on a
toboggan after consideration of the force of gravity, friction, and
aA.
A33
found that a worker can p,roduce at most 30 units per day.
The number of units nf per day pro&rced by a new employee will increase at a rate proportional to the difference
between 30 and M This is described by the differential
equation
v:
Slop:y':4
DifferentialEquations
Learning Theory The manageme,nt of a f*ctory
dN
34. Point (8,2)
E
u*n-
Solve this differential equation.
Fqmiilgs (hdoflars)