diff eqn differential equations
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A20 APPENDIX C Differential Equations
General Solution of a Differential Equation
A differential equation is an equation involving a differentiable function and one
or more of its derivatives. For instance,
Differential equation
is a differential equation. A function is a solution of a differential equa-
tion if the equation is satisfied when y and its derivatives are replaced by and
its derivatives. For instance,
Solution of differential equation
is a solution of the differential equation shown above. To see this, substitute for y
and in the original equation.
Substitute for y and
In the same way, you can show that and are
also solutions of the differential equation. In fact, each function given by
General solution
where C is a real number, is a solution of the equation. This family of solutions
is called the general solution of the differential equation.
EXAMPLE 1 Checking Solutions
Show that (a) and (b) are solutions of the differential equation
Solution
(a) Because and it follows that
So, is a solution.
(b) Because and it follows that
So, is also a solution. y Ce x
y y Ce x Ce x
0.
y Ce x, y Ce x
y Ce x
y y Ce x Ce x
0.
y Ce x, y Ce x
y y 0.
y Ce x y Ce x
y Ce2 x
y 12e2 x y 2e2 x, y 3e2 x,
y . y 2 y 2e2 x 2e2 x 0
y 2e2 x
y e2 x
f x y f x
y 2 y 0
C Differential Equations
C.1 Solutions of Differential Equations
Find general solutions of differential equations. • Find particular solutions of differential equations.
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Particular Solutions and Initial Conditions
A particular solution of a differential equation is any solution that is obtained
by assigning specific values to the constants in the general equation.*
Geometrically, the general solution of a differential equation is a family of
graphs called solution curves. For instance, the general solution of the differen-tial equation is
General solution
Figure A.7 shows several solution curves of this differential equation.
Particular solutions of a differential equation are obtained from initial
conditions placed on the unknown function and its derivatives. For instance,
in Figure A.7, suppose you want to find the particular solution whose graph
passes through the point This initial condition can be written as
when Initial condition
Substituting these values into the general solution produces which
implies that So, the particular solution isParticular solution
EXAMPLE 2 Finding a Particular Solution
Verify that
General solution
is a solution of the differential equation for any value of C . Then
find the particular solution determined by the initial condition
when Initial condition
Solution The derivative of is Substituting into the differen-tial equation produces
Thus, is a solution for any value of C . To find the particular solution,
substitute and into the general solution to obtain
or
This implies that the particular solution is
Particular solution
*Some differential equations have solutions other than those given by their general solutions. These
are called singular solutions. In this brief discussion of differential equations, singular solutions will
not be discussed.
y 2
27
x3.
C 2
27.2 C 33
y 2 x 3
y Cx3
0.
xy 3 y x3Cx2 3Cx3
y 3Cx
2
. y Cx3
x 3. y 2
xy 3 y 0
y Cx3
y 3 x2.C
3.
3 C 12,
x 1. y 3
1, 3.
y Cx2.
xy 2 y 0
APPENDIX C Differential Equations A21
3
2
3
2
1
3 2 32
y
x
2 xC y) ) ,1 3
FIGURE A.7
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EXAMPLE 3 Finding a Particular Solution
You are working in the marketing department of a company that is producing a
new cereal product to be sold nationally. You determine that a maximum of 10
million units of the product could be sold in a year. You hypothesize that the rateof growth of the sales x (in millions of units) is proportional to the difference
between the maximum sales and the current sales. As a differential equation, this
hypothesis can be written as
The general solution of this differential equation is
General solution
where t is the time in years. After 1 year, 250,000 units have been sold. Sketch
the graph of the sales function over a 10-year period.
Solution Because the product is new, you can assume that when So,
you have two initial conditions.
when First initial condition
when Second initial condition
Substituting the first initial condition into the general solution produces
which implies that Substituting the second initial condition into the
general solution produces
which implies that So, the particular solution is
Particular solution
The table shows the annual sales during the first 10 years, and the graph of the
solution is shown in Figure A.8.
In the first three examples in this section, each solution was given in
explicit form, such as Sometimes you will encounter solutions for
which it is more convenient to write the solution in implicit form, as illustrated
in Example 4.
y f x.
x 10 10e0.0253t .
k ln4039 0.0253.
0.25 10 10ek (1)
C 10.
0 10 Cek (0)
t 1 x 0.25
t 0 x 0
t 0. x 0
x
10
Cekt
0 ≤ x ≤ 10.dx
dt k 10 x,
A22 APPENDIX C Differential Equations
Sales Projection
Sales(inmillionsofunits)
Time (in years)
3
2
1
x
t
1 2 3 4 5 6 7 8 9 10
x e= 10 10− −0.0253t
FIGURE A.8
t 1 2 3 4 5 6 7 8 9 10
x 0.25 0.49 0.73 0.96 1.19 1.41 1.62 1.83 2.04 2.24
Rate of change
of x
is propor-tional to
the differencebetween10 and x.
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EXAMPLE 4 Sketching Graphs of Solutions
Verify that
General solution
is a solution of the differential equation Then sketch the particular
solutions represented by and
Solution To verify the given solution, differentiate each side with respect to x.
Given general solution
Differentiate with respect to x.
Divide each side by 2.
Because the third equation is the given differential equation, you can conclude
that
is a solution. The particular solutions represented by andare shown in Figure A.9.C ±4
C ±1,C 0,
2 y2 x2
C
2 yy x 0
4 yy 2 x 0
2 y2 x2
C
C ±4.C 0, C ±1,
2 yy x 0.
2 y2 x2
C
APPENDIX C Differential Equations A23
22
2
3
2
2
1
2
3
1
y
x
x
y
x
y
x
x
C 0
1C
1C
C 4
C 4
y y
TAKE ANOTHER LOOK
Writing a Differential Equation
Write a differential equation that has the family of circles
as a general solution.
x 2 y 2 C
FIGURE A.9 Graphs of Five Particular Solutions
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The following warm-up exercises involve skills that were covered in earlier sections.
You will use these skills in the exercise set for this section.
In Exercises 1–4, find the first and second derivatives of the function.
1.
2.
3.
4.
In Exercises 5–8, use implicit differentiation to find
5.
6.
7.
8.
In Exercises 9 and 10, solve for k .
9.
10. 14.75 25 25e2k
0.5 9 9ek
3 xy x2 y 2 10
xy2 3
2 x y3 4 y
x2 y 2
2 x
dy dx .
y 3e x2
y 3e2 x
y 2 x3 8 x 4
y 3 x2 2 x 1
A24 APPENDIX C Differential Equations
EXER CI SES C.1
In Exercises 1–10, verify that the function is a solution of the
differential equation.
Solution Differential Equation
1.
2.3.
4.
5.
6.
7.
8.
9.
10.
In Exercises 11–28, verify that the function is a solution of the
differential equation for any value of C .
Solution Differential Equation
11.
12.
13.
14.
15.
16.
17.
18.
19.
20. y y 0 y C 1 C 2e x
xy y x3 x 4 y x2 2 x
C
x
y y x 2 x y x ln x2 2 x32 Cx
xy 3 x 2 y 0 y Cx2 3 x
y y 10 0 y Cet 10
3dy
dt y 7 0 y Cet 3
7
dy
dx 4 y y Ce4 x
dy
dx 4 y y Ce4 x
dy
dx
x
4 x2 y 4 x2
C
dy
dx
1
x2 y
1
x C
y 3 x2 y 6 xy 0 y e x3
y y 2 y 0 y 2e2 x
xy 2 y 0 y 1
x
x2 y 2 y 0 y x2
y 2
x y 0 y 4 x2
y 3
x y 0 y 2 x3
y 2 xy 0 y 3e x2
y 2 y 0 y e2 x
y 6 x
2
1 y 2 x3
x 1
y 3 x2 y x3 5
WA R M -UP C.1
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Solution Differential Equation
21.
22.
23.
24.
25.
26.
27.
28.
In Exercises 29–32, use implicit differentiation to verify that the
equation is a solution of the differential equation for any value
of C .
Solution Differential Equation
29.
30.
31.
32.
In Exercises 33–36, determine whether the function is a solution
of the differential equation
33.
34.
35.
36.
In Exercises 37–40, determine whether the function is a solution
of the differential equation
37.
38.
39.
40.
In Exercises 41–48, verify that the general solution satisfies the
differential equation.Then find the particular solution that satis-
fies the initial condition.
41. General solution:
Differential equation:
Initial condition: when
42. General solution:
Differential equation:
Initial condition: when
43. General solution:
Differential equation:
Initial condition: and when
44. General solution:
Differential equation:Initial condition: and when
45. General solution:
Differential equation:
Initial condition: and when
46. General solution:
Differential equation:
Initial condition: when
47. General solution:
Differential equation:
Initial condition: when
when
48. General solution:
Differential equation:Initial condition: and when
In Exercises 49–52, the general solution of the differential
equation is given. Use a graphing utility to graph the particular
solutions that correspond to the indicated values of C .
General Solution Differential Equation C-values
49. 1, 2, 4
50. 0,
51.
52.
In Exercises 53–60, use integration to find the general solution of
the differential equation.
53.
54.
55.
56.
57.
58.
59.
60. dy
dx xe x
dy
dx x x 3
dydx
x1 x2
dy
dx
1
x2 1
dy
dx
x 2
x
dy
dx
x 3
x
dy
dx
1
1 x
dy
dx 3 x2
0, ±1, ±2 y y 0 y Ce x
0, ±1, ±2 x 2 y 2 y 0 y C x 22
±1, ±44 yy x 04 y 2 x2
C
xy 2 y 0 y Cx2
x 0 y 1 y 2 y 4 y 4 y x2e2 x
y C 1 C 2 x 112 x
4e2 x
x 3 y 0
x 0 y 4
9 y 12 y 4 y 0
y e2 x3C 1 C 2 x
x 1 y 2
y 2 x 1 y 0
y Ce x x 2
x 0 y 6 y 5
y y 12 y 0
y C 1e4 x C 2e
3 x
x 2 y 4 y 0 x2
y
3 xy
3 y
0
y C 1 x C 2 x3
x 1 y 0.5 y 5
xy y 0
x > 0 y C 1 C 2 ln x,
x 1 y 2
2 x 3 yy 0
2 x2 3 y 2
C
x 0 y 3
y 2 y 0
y Ce2 x
y x ln x
y xe x
y 4e x
29 xe2 x
y 29 xe2 x
y 3 y 2 y 0.
y 4e2 x
y 4 x
y 5 ln x
y e2 x
y 4 16 y 0.
y3 y x2 y 2
0 x2 y 2
C
x2 y 2 x y 0 x2 xy C
x y y x y 0 y 2 2 xy x2
C
y 2 xy
x2 y 2
x2 y 2
Cy
x y xy 0 y xln x C
x y 1 y 4 0 y x ln x Cx 4
y 2 x 1 y 0 y Ce x x2
y 2 xy xy2 y 2
1 Ce x2
2 xy y x3 x y x3
5 x C x
y
ay
x bx3
y
bx4
4 a Cxa
y 3 y 4 y 0 y C 1e4 x C 2e
x
2 y 3 y 2 y 0 y C 1e x2
C 2e2 x
APPENDIX C Differential Equations A25
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A26 APPENDIX C Differential Equations
In Exercises 61–64,you are shown the graphs of some of the solu-
tions of the differential equation. Find the particular solution
whose graph passes through the indicated point.
61. 62.
63. 64.
65. Biology The limiting capacity of the habitat of a wildlife
herd is 750. The growth rate of the herd is propor-
tional to the unutilized opportunity for growth, as described
by the differential equation
The general solution of this differential equation is
When the population of the herd is 100. After 2
years, the population has grown to 160.
(a) Write the population function N as a function of t .
(b) Use a graphing utility to graph the population function.
(c) What is the population of the herd after 4 years?
66. Investment The rate of growth of an investment is
proportional to the amount in the investment at any time t .
That is,
The initial investment is $1000, and after 10 years the bal-
ance is $3320.12. The general solution is
What is the particular solution?
67. Marketing You are working in the marketing depart-
ment of a computer software company. Your marketing
team determines that a maximum of 30,000 units of a new
product can be sold in a year. You hypothesize that the rate
of growth of the sales x is proportional to the difference
between the maximum sales and the current sales. That is,
The general solution of this differential equation is
where t is the time in years. During the first year, 2000
units are sold. Complete the table showing the numbers of
units sold in subsequent years.
68. Marketing In Exercise 67, suppose that the maximum
annual sales are 50,000 units. How does this change the
sales shown in the table?
69. Safety Assume that the rate of change in the number of
miles s of road cleared per hour by a snowplow is inverse-
ly proportional to the depth h of the snow. This rate of
change is described by the differential equation
Show that
is a solution of this differential equation.
70. Show that is a solution of the differen-
tial equation
where k is a constant.
71. The function is a solution of the differential
equation
Is it possible to determine C or k from the informationgiven? If so, find its value.
True or False? In Exercises 72 and 73, determine whether
the statement is true or false. If it is false, explain why or give an
example that shows it is false.
72. A differential equation can have more than one solution.
73. If is a solution of a differential equation, then
is also a solution. y f x C
y f x
dy
dx 0.07 y.
y Cekx
y a b y a 1
k dy
dt
y a Cek 1bt
s 25 13
ln 3
lnh
2
ds
dh
k
h.
x 30,000 Cekt
dx
dt k 30,000 x.
A Cekt .
dA
dt kA.
t 0,
N 750 Cekt .
dN dt k 750 N .
dN dt
1
2
1
2
x
(2, 1)
y
1 2 3123
4
5
6
x
(0, 3)
y
2 xy y 0 y y 0
y 2 2Cx y Ce x
4
3
2
4
3
433 x
)4,3(
y
7
4
3
2
1
4
3
2
1
654 x
)4,4(
y
yy 2 x 02 xy
3 y 0
2 x2 y 2
C y 2 Cx3
Year, t 2 4 6 8 10
Units, x