Differential Equations for Engineers and Scientists by Y. Cengel and W. Palm III
ANSWERS TO SELECTED PROBLEMS PROBLEMS ( Answers to Section Review problems are in the textbook )
CHAPTER 1
1-33 1-35 1-37
with
with
,
1-41C The slope at the given point is . 1-43C This 1-43C This can be possible if
or
, or both are zero.
1-45C High pressure lines are steeper than the low pressure lines. 1-47
1-51
is a continuous function on
(b) (c)
is defined on and continuous in that interval is continuous for all except for
(d)
1-49
(a)
is continuous on
constant
(a)
, where denotes set of reel numbers
satisfies the given condition
(b)
satisfies the given condition
(c) No elementary function can satisfy the given condition.
1-53
(a) Given: Solution: (b) Given:
CHAPTER 1
1-33 1-35 1-37
with
with
,
1-41C The slope at the given point is . 1-43C This 1-43C This can be possible if
or
, or both are zero.
1-45C High pressure lines are steeper than the low pressure lines. 1-47
1-51
is a continuous function on
(b) (c)
is defined on and continuous in that interval is continuous for all except for
(d)
1-49
(a)
is continuous on
constant
(a)
, where denotes set of reel numbers
satisfies the given condition
(b)
satisfies the given condition
(c) No elementary function can satisfy the given condition.
1-53
(a) Given: Solution: (b) Given:
Solution:
(c) Given: Solution: 1-55
(a) Given: Solution:
(b) Given: Solution: c) Given:
Solution:
Solution:
d) Given:
1-57
Solution:
(b) Given:
Solution: 1-59
(a) Given: Solution:
Solution:
(d) Given:
Solution:
(c) Given:
1-63
(b) Given: Solution:
(a) Given:
(a) (b)
(Linear, constant coefficient) (Linear, variable coefficient) (c) (Linear, variable coefficient) (d) (Linear, constant coefficient)
(e)
1-69 1-71 1-73 1-75 1-77 1-79
(Nonlinear, variable coefficient)
and are the solutions of the differential equation. and are the solutions of the differential equation. and are the solutions of the differential equation. and are the solutions of the differential equation.
with and
(a) (b)
, where
(Upward direction is positive)
and are arbitrary constants
cannot be solved by direct integration , where and are arbitrary constants. (c) (d) cannot be solved by direct integration (e) The unknown function cannot be found in terms of elementary functions.
1-81
, where is an arbitrary constant. (b) cannot be solved by direct , where and are arbitrary constants (c) (d) cannot be solved by direct integration (e) The unknown function cannot be found in terms of elementary functions. (a)
1-83C The ginput function gives the result
rad.
1-87 The result is 0.4304077247. 1-89
(a) The answer is
If . (b) The answer is 1-91
(a) The answer is
(b) The answer is
if , and
or
Another form returned is
(c) The answer is
(d) The answer is
or
(b) (c) (d)
1-95
1-97
,
(a) (b) (c) (a) (b)
and .
.
and
.
and
.
.
(c)
1-99
.
(a)
(b)
.
(c)
1-101 (a)
(b)
1-103 (a) (b) 1-105
.
.
m/s.
1-107
and
1-109
1-111 (a)
and
and (b)
1-93 (a)
.
CHAPTER 2
2-37
(a)
2-39
(a)
linear
(b)
nonlinear
2-41 (a)
(b)
nonlinear
nonlinear
, (b)
2-43 (a)
where
, (b)
2-45 (a)
where
.
(b)
2-47 (a)
, (b)
2-49 (a)
, (b)
2-51 2-53
is also a solution of
, no matter what value of
2-55
cannot be a solution of the given DE.
is.
2-59
2-63 (a) At steady state, y = 24/3 = 8. (b) 5.04 is 63% of 8, so it will take one time constant, or 8 , to reach 5.04. (c) 7.84 is 98% of 8, so it will take four time constants, or 32 , to reach 7.84. 2-65
.
2-67
2-69 the bottom of the pond is 2-71 2-73
2-75
. Therefore we conclude that the fraction of the light that will reach .
The amount of salt after 30 minutes will be , The solution is
, and it will never drop to
.
, The terminal velocity is
2-77
.
2-79 a) Theorem 2-2 guarantees both existence and uniqueness of a solution in a neighborhood of any . b) The given differential equation must have a unique solution near any point in the -plane where or . 2-81 a) b) 2-83 a) b)
Theorem 2-2 guarantees both existence and uniqueness in some neighborhood of Theorem 2-2 guarantees both existence and uniqueness in some neighborhood of
2-87 (a)
, (b)
2-89 (a) 2-91
Theorem 2-2 guarantees nothing in some neighborhood of . The Theorem 2-2 guarantees both existence and uniqueness in some neighborhood of
.
. .
, (b)
, where
(a)
.
(b)
2-93 (a)
, (b)
2-95 (a)
, (b)
2-97 (a)
, (b)
2-99
, The time required for the tank to be empty can be evaluated by
setting
which yields
2-101
, where
Equilibrium points are 2-103 (a)
,
, and
.
,
, where
(b)
2-105 (a)
2-109 (a) 2-111 (a)
, (b)
is homogeneous, (b)
is homogeneous, (b)
is not homogeneous . is not homogeneous.
2-113 (a)
, (b)
, where
.
, where
2-115 (a)
(b)
2-117 (a) 2-119 (a)
, (b)
, where
.
, (b)
2-121 (a)
,
(b)
2-123 (a)
, (b)
2-127 (a)
, where
(b) The differential equation is inexact
2-129 (a)
, (b) The differential equation is inexact.
2-131 (a) 2-133 (a)
, (b) The differential equation is inexact.
, (b) The differential equation is inexact.
2-135 2-137 2-139 2-147
2-149 Maple gives the answer:
2-151 The equation in the first printing is incorrect. It should be
.The solution is
2-159 2-161 2-163
2-165 (a)
, (b)
(c)
(downwards).
2-167 2-169
Taking
we end up with the
whose limit is as
.
2-171 2-173 2-175 2-177 2-179 2-181 2-183 2-185 2-187 2-189 2-191 2-193 2-195 2-197 2-199 2-201 2-203 2-207 2-209
.
3-57 (a) (b) (c)
CHAPTER 3
; Nonlinear, nonhomogeneous, constant coefficients ; Linear, homogeneous, constant coefficients ; Linear, homogeneous, variable coefficients
(d)
; Linear, nonhomogeneous, variable coefficients
3-59 (a)
; Nonlinear, nonhomogeneous, constant coefficients
(b) ; Noninear, homogeneous, constant coefficients (c) ; Linear, homogeneous, variable coefficients (d) ; Linear, nonhomogeneous, constant coefficients 3-61(a) The initial-value problem has a unique solution in the interval (b) The initial-value problem has a unique solution in the interval 3-63 (a)The initial-value problem has a unique solution in the interval (b)The initial-value problem has a unique solution in the interval
3-65 (a)
, (b)
3-67 (a)
.
.
.
.
, (b)
3-73 (a)
and
are linearly dependent, (b)
3-75 (a)
and
are linearly independent, (b)
and
are linearly independent.
3-77 (a)
and
are linearly independent, (b)
and
are linearly independent.
3-79 (a)
and
are linearly dependent, (b)
3-81
and
are linearly dependent.
and
are linearly independent.
and
are linearly independent.
and
are linearly dependent.
and
are linearly independent.
3-83 3-85 3-87 3-89
3-93 (a)
is also
a solution, (b)
and
and
is not a solution
(c)
is not a solution, (d)
is not a solution
3-95 (a)
is not a solution, (b)
is not a solution
(c)
is also
a solution, (d)
is also
a solution
are linearly dependent.
are linearly dependent.
3-97 (a) (c) 3-99 (a) (c)
is also
is not a solution, (d) is not a solution, (b) is also
a solution, (b)
is not a solution
a solution, (d)
3-101 (a) The Wronskian of
and
(b) The Wronskian of
and
(c) The Wronskian of
and
3-103 (a) The Wronskian of
and
(b) The Wronskian of
and
(c) The Wronskian of
and
is not a solution
is also
a solution
is also
a solution
is never zero for
is zero
is zero is never zero for is zero
is zero
3-105 (a) (b) (c)
and
does not form a fundamental set of solutions.
3-107 (a) (b) (c) 3-111 3-113 3-115 3-117 3-119 3-121
3-129 (a) 3-131 (a) (c) 3-133 3-134
, (b)
, (b)
, (c)
,
3-135 3-137 3-139 3-141 3-143
3-147 (a)
, (b)
3-149 (a)
(b)
3-151 (a) 3-153 3-155
, (b)
3-161 (a)
, (b)
(c)
, (d)
3-163 (a)
, (b)
(c)
(d)
3-165 (a) (b) (c) (d) 3-167 3-169
3-171 (a) (b)
3-173 (a) (b)
3-175 (a)
(b)
3-177 (a)
, (b)
3-181 (a)
, (b)
3-183 (a)
, (b)
3-185 (a) (b) 3-193 3-195 3-197
,
,
,
,
will cause the resonance.
,
3-199 The mass will pass through its static equilibrium position at the time velocity of 3-201 3-203 3-205 3-207 3-209 3-211 3-215
, with a
,
0.2080 m
The charge of capacitor would be, at least mathematically , unbounded as 3-217 If
then there are two real and distinct roots,
the differential equation is
and
. Thus the general solution of
If
then there are two real and equal roots,
. Thus the general
solution of the differential equation is
If
then there are two complex and conjugate roots,
. Thus the general
solution of the differential equation is
where
and
3-219 3-221 3-223 3-225 3-227 3-229 3-230 3-231 3-233 3-235
3-237 Note: the equation is incorrect in the first printing of the textbook. It should be . The solution is
3-241 3-243 3-245 3-247 3-249
3-251 3-253 3-255 3-257 3-259 3-261 3-263 3-265 3-267 3-269
,
.
,
4-25 (a)
(b) (c) (d) 4-27 (a) (b) (c) (d)
CHAPTER 4 ; Nonlinear, nonhomogeneous, constant coefficients
; Linear, homogeneous, constant coefficients ; Linear, homogeneous, variable coefficients
; Linear, nonhomogeneous, variable coefficients
; Nonlinear, nonhomogeneous, constant coefficients Nonlinear, homogeneous, constant coefficients ; Linear, homogeneous, variable coefficients
; Linear, nonhomogeneous, constant coefficients
4-29 (a) The initial-value problem has a unique solution in the interval (b) The initial-value problem has a unique solution in the interval 4-31 (a) The initial-value problem has a unique solution in the interval (b) The initial-value problem has a unique solution in the interval
4-35 (a) The Wronskian of these three solution functions is never zero for (b) The solutions 4-37 (a)
,
(b)
,
,
and
and
and
are linearly dependent.
are linearly independent.
(b)
and
are linearly independent.
4-53 (a) 4-55 (a) (b)
4-57 4-59
.
.
.
do not form a set of fundamental solutions.
and
4-45
.
do not form a set of fundamental solutions.
4-39 (a)
4-43
.
, (b) Given:
,
4-61 4-63
4-67 (a) (b)
(c)
(d)
4-69 (a)
(b)
(c)
(d)
4-71 (a) (b)
(c)
(d) 4-73
(
4-77 (a)
(b)
4-79 (a) (b)
4-81 (a) (b) 4-87 4-89
,
4-91
4-93 (a) (b) (c) (d) (e) (f) (g) 4-95 (a) (b)
4-97 4-99 (a) 4-103 4-105 4-107 4-109 4-111 4-113 4-115 4-117 4-119 4-121 4-123 4-125
, (b)
CHAPTER 5
5 – 41 (a)
(b) 5 – 43 (a)
, (b)
5 – 45 (a)
, (b)
5 – 47 (a) 5 – 49
, (b)
5 – 51 The equality holds for any value. 5 – 53 The equality holds for any value. 5 – 55 Not 55 Not correct 5 – 57 (a)
,
, (b)
5 – 59 (a)
,
, (b)
5 – 61 (a)
, (b)
5 – 63 (a)
(b) 5 – 65
5 – 69 (a) All points are ordinary points. (b) Both
and
are the regular singular points of the differential equation.
5 – 71 (a) Both
and
are the regular singular points of the differential equation.
(b) All points are ordinary points of the differential equation. 5 – 73 5 – 75 5 – 77 5 – 79
. . .
Interval of convergence:
.
5 – 81
5 – 83
Interval of the convergence:
.
Interval of the convergence:
.
5 – 85 5 – 87
5 – 89
Interval of convergence is
.
5 – 91 5 – 97
,
.
5 – 99
5 – 105 (a)
(b)
5 – 107 (a) (b)
5 – 109 (a) , where and . Since is the only singular point for the given differential equation, the series solution converges for all . (b) , where . It is clear from either or that is another singular point of the given differential equation. Therefore the series will converge for all such that . 5 – 111 (a) where and whereas the constant may be zero. The series solution will converge for any . (b) where , whereas the constant may be zero. The series solution will converge for any .
,
and
5 – 113 (a) , where and . Since is the only singular point for the given differential equation, the series solution converges for all . (b) , where and , whereas the constant may be zero. It is clear from either or that are two other singular points of the given differential equation. Therefore the series solution will converge for all such that . 5 – 115 (a)
(b)
5 – 117 (a)
(b)
5 – 119 (a)
, (b)
5-125 5-129 (a)
, The integral in the result cannot be evaluated in finite form in
term f y f te kw Bee’ fut (b)
5-131 (a)
The series solution found in Problem 5-63a is
(b)
The series solution found in Problem 5-63b is
5-133
The series solution found in Problem 5-65 is
5-135
The series solution found in Problem 5-80 is
5-137
The series solution found in Problem 5-82 is
5-139
The series solution found in Problem 5-84 is
5-141
The series solution found in Problem 5-86 is
5-143
The series solution found in Problem 5-88 is
5-145
The general series solution found in Problem 5-90 is
The assumed power series solution suggests that and . Therefore the solution of the given initial-value problem can be acquired by simply plugging in and in the general solution. Then the solution of the initial-value problem is . 5-147 The general series solution found in Problem 5-92 is
The assumed power series solution suggests that and . Therefore the solution of the given initial-value problem can be acquired by simply plugging in and in the general solution. Then the solution of the initial-value problem is obtained to be
5-149 (a)
The solution found in Problem 5-115(a) is
(b)
or using MuPAD
The solutions found from Maple and MuPAD differ, but they are both correct solutions. The solution
found in Problem 5-115(b) is
5-151 (a) (b)
Maple is unable to solve this problem. The solution found in Problem 5-117(b) is
5-153 (a) (b)
, The solution found in Problem 5-119(a) is ,The solution found in Problem 5-119(b) is
CHAPTER 6 6-21 (a)
(b)
6-23 (a)
(b)
6-25 (a)
(b)
6-27
6-29
6-31 The system is nonlinear due to term coefficients due to term
, nonhomogeneous due to
, and has variable
.
6-33 The system is nonlinear due to terms variable coefficients due to term
and
, nonhomogeneous due to
, and has
.
6-35 The system is linear, nonhomogeneous due to terms
and , and has constant coefficients.
6-37 The system is linear, nonhomogeneous due to , and has variable coefficients due to terms and
6-39
.
6-41
6-43
6-45
6-47 (a)
(b)
6-49 (a)
(b)
6-51 (a)
(b)
6-53 (a)
(b)
6-55 (a)
(b)
6-57 (a)
(b)
6-59 (a)
(b)
6-61 (a)
(b)
6-63
6-65
6-67 6-69
6-73 (a)
(b)
6-75 (a)
and
(b)
6-77 (a)
(b)
6-79 (a)
(b)
6-81 (a)
(b)
6-83
6-85
6-87
6-91
6-93
6-95 (a)
(b)
6-97
x t
389 95
y t
5841 95
x t
1
32110 64220
1 373 3/2t 1 35 3 95t e cos 95t t 2 338 2 338 13
e3/ 2t sin
1 127 3/ 2t 1 127 1 95t e cos 95t t 2 676 2 676 26
e3/ 2t sin
6-99
y t
6-101
287 71 1 17 1 sin 71t cos 71t e1/2 t 9 2 9 2 639 517 71 1 59 1 sin 71t 71 cos 71t e1/2 t 18 18 2 2 1278 5
6-103
CHAPTER 7
7-39 (a)
(c)
, (b)
, (d)
7-41 (a)
(c)
, (b)
,
, (d)
7-43
(a)
c)
, (b)
, (d)
7-45 (a)
b)
c)
,
,
,
7-47 (a)
,
b)
c)
,
,
7-49 (a)
(b)
7-51 (a)
(b)
(c)
(d)
7-53
m1 x1 k1 x1 k2 ( x2 x1 ) c1 x1 c2 ( x2 x1) m2 x2 f k2 ( x2 x1 ) c2 ( x2 x1) 0 0 k2 k1 A m1 k2 m2
7-65 (a)
(b)
7-67 (a)
(b)
0
1
0
0
k2 m1
c1
c2 m1
k2
c2
m2
m2
1 c2 m1 c 2 m2 0
0 0 B 0 1
7-69 (a)
(b)
7-71 (a)
(b)
7-73 (a)
(b) The inverse of the square matrix
does not exist. This is a singular matrix.
7-75 (This problem is identical with 7-14, and will be removed in the second press run) (a)
(b)
(c)
(d)
7-77 (a)
(b)
(c) The system has no solution.
(d)
7-79
(a)
(b)
(c)
(d)
7-81 The vectors are linearly independent. 7-83 The vectors are linearly independent. 7-85 The vectors are linearly independent.
7-87 The vectors are linearly independent in the given interval. 7-89 The vectors are linearly dependent in 7-91 (a)
(b)
and
,
and
.
7-93 (a)
(b)
and
and
7-95 (a)
(b)
and
,
and
7-97 (a)
(b)
,
,
and
7-99 (a)
(b)
,
,
and
7-105
and
are not solutions to the given system, and they are linearly independent.
7-107
and
are the solutions to the given system, and they are linearly dependent.
7-109 and are the solutions to the given system, and they are linearly independent. Thus, the general solution of the given system is
7-111 7-113
,
and
are not solutions to the given system, and they are linearly dependent.
,
and
are the solutions to the given system, and they are linearly dependent
7-115 The vector
satisfies the given system, and it is a solution.
7-117 The vector
does not satisfy the given system, and it is not a particular solution.
7-119 The vector
does not satisfy the given system.
7-125
7-127
7-129
7-131
7-133
7-135
7-137
7-139
7-147
7-149
7-151
7-153
7-155
7-157
7-159
7-161
7-163
7-165
( a) A
3
63 62 62 63
1
(b) A I A
1
(6I A) / 5
7-167 Only the second mode is controllable.
1
2 1 3 3 5 2
7-169 The truncated series solution gives whereas from the example
A
7-171
( a)
(c)
, (b)
,
, (d)
7-173
(a)
A
5 3 0 4
5
(b) A
1
3
4
B
B
0 5
4 0 0 5
C
1 0
C
1
0
1
0
D
D
0 0
0
0
7-181
7-183
7-185
7-187
7-189
7-191
7-193
7-195
Note that the initial conditions specified for the given system of two linear homogeneous differential equations with constant coefficients are both equal to zero. Therefore this initial-value problem has only the trivial solutions .
7-197
7-199
7-201
CHAPTER 8
8-17 (a) 8-19 (a)
, (b)
, (c)
(b)
(c)
8-27 (a) (c)
8-37 (a)
, (b)
, (d)
(b)
8-39 (a) 8-41 (a)
8-49 (a) 8-51 (a) 8-53 (a)
(c)
(b)
(c)
(b)
(c)
(b)
(b)
(b)
8-55 (a) 8-57 8-59 8-61
(b)
8-65 (a)
(b)
8-67 (a)
(b)
8-71 (a) 8-73 (a)
(b)
(b)
8-75 (a) 8-81 (a)
(c)
(b)
(c)
(b)
8-83 (a) 8-85 (a)
(c)
(b)
(b)
8-91 (a) 8-93 (a)
8-95 (a) 8-97 (a) 8-101 8-103 8-105 8-107 8-109
8-111 8-113 8-115 8-117 8-119 8-121
8-123
8-125
(b)
(b)
(b)
(b)
8-127
8-129 8-131 8-133 8-135
8-137 (a)
1
5t
45e
18e
2 t
1
30
65e 2t
65
7
(c)
8-139 (a)
1
2sin 3t cos 3 t 3 1
(b)
(d)
24e3t 2 3e2t
1 9e
2t
6e3t
9e5t
1
40e
30
5t
7
1
13
13
1
30e
15
5t
(b)
(c)
(d)
8-147 F ( s)
8-149 x(t )
C Ds
2 15
2
t5
2 3
C Ds
2
e Ds
C s
eDs
t
t 4 3t 3 9t 2 19t 19 19e
8-151 8-153
8-155 (a) p0 30 103 Pa
0.2/ ln0.5 0.289
(b) x(t ) 0.643cos10t 0.2225sin10t 0.643e 8-157 x
a b 2
8-159 x(t )
8-161
2 n
F0 kT
b
t
F0 k
n
3.46t
sin n t n2 cosn t n2 ebt
F0 kT n
sinn t
F 0
cos nt k
CHAPTER 9 9-29
e
Trapezoidal rule, N=1:
9-31
e ,
,
, Trapezoidal rule, N=2:
Simpson’s rule, N=1
Strip method, N=1:
Strip method, N=1:
,
, Simpson’s rule, N=2:
,
, Strip method, N=2:
Simpson’s rule, N=1 and 2:
9-35
ee ,
, Trapezoidal rule, N=2:
Trapezoidal rule, N=1:
9-33
, Strip method, N=2: e e ,
Strip method, N=1: ,
Trapezoidal rule, N=1: Simpson’s rule, N=1:
,
,
.
,
, Strip method, N=2:
,
,
.
, Trapezoidal rule, N=2:
,
, Simpson’s rule, N=2:
,
,
9-37 Exact results and all numerical methods result in 0. Relative error 0.00%. 9-39 Strip Method for N=10: Strip Method for N=100: Trapezoidal Rule for N=10: Trapezoidal Rule for N=100: 9-41 Trapezoidal Rule for N=10: Trapezoidal Rule for N=100: Strip Method for N=10: Strip Method for N=100: Simpson’s Rule for N=10: Simpson’s Rule for N=100:
.
.
.
.
9-43 Strip Method for N=10: Strip Method for N=100: Simpson’s Rule for N=10 and N=100:
9-45 Strip Method for N=10: Strip Method for N=100: Trapezoidal Rule for N=10:
Trapezoidal Rule for N=100:
Simpson’s Rule for N=10:
Simpson’s Rule for N=100:
9-53 Trapezoidal Rule a) One step:
,
b) Two steps: Simpson’s Rule
a) One step:
b) Two steps: 9-55 Strip Method a) One step:
,
,
,
,
b) Two steps: Simpson’s Rule
a) One step:
b) Two steps:
,
,
,
9-57 Strip Method a) One step:
,
b) Two steps:
,
Trapezoidal Rule a) One step:
b) Two steps: Simpson’s Rule
a) One step:
,
,
,
b) Two steps: 9-59 Strip Method
After 10 steps with h=0.2: After 20 steps with h=0.1:
Trapezoidal Rule After 10 steps with h=0.2: After 20 steps with h=0.1: 9-61 Trapezoidal Rule After 10 steps with h=0.2: After 20 steps with h=0.1: Simpson’s Rule
After 10 steps with h=0.2: After 20 steps with h=0.1:
9-63
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.
,
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.
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Strip Method
After 10 steps with h=0.2:
,
.
After 20 steps with h=0.1:
,
.
Simpson’s Rule
After 10 steps with h=0.2:
,
After 20 steps with h=0.1:
, 0.00%
9-65 Strip Method
After 10 steps with h=0.2:
,
After 20 steps with h=0.1:
,
Trapezoidal Rule
After 10 steps with h=0.2:
,
After 20 steps with h=0.1:
,
Simpson’s Rule
After 10 steps with h=0.2:
,
After 20 steps with h=0.1:
,
9-71 After one step:
After two steps:
9-75 After one step: ,
After two steps: ,
9-79
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.
.
After two steps:
9-73 After one step: ,
9-77 After one step:
.
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After two steps:
After 10 steps with h=0.2:
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.
After 20 steps with h=0.1:
,
.
,
9-81 There is a typographical error in this problem. The change is yellowed below and will be corrected in the second printing. Given: , After 10 steps with h=0.2: After 20 steps with h=0.1:
,
,
.
9-83
After 10 steps with h=0.2:
,
After 20 steps with h=0.1:
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.
.
9-85 After 10 and 20 steps with h=0.2 and h=0.1:
,
.
9-87 There is a typographical error in this problem. The change is yellowed below and will be , corrected in the second printing. Given: After 10 steps with h=0.2:
,
.
After 20 steps with h=0.1:
,
.
9-89
After 10 steps with h=0.2:
,
.
After 20 steps with h=0.1:
,
.
9-91 There is a typographical error in this problem. The change is yellowed below and will be corrected in the second printing. Given:
,
After 10 steps with h=0.2:
,
.
After 20 steps with h=0.1:
,
.
9-93
After 10 steps with h=0.2:
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.
After 20 steps with h=0.1:
,
.
9-105 There is a typographical error in this problem. The change is yellowed below and will be corrected in the second printing.Given: After one step: :
After two steps: 9-107 After one step:
After two steps:
9-109
,
,
(a)
2.00000 -0.77784 -0.77878 -0.77915
0.04788
0.16917
(b)
2.00000 -0.77822 -0.77908 -0.77915
0.00964
0.11976
9-111 There is a typographical error in this problem. The change is yellowed below and will be corrected in the second printing. Given: (a)
2.00000 -1.64591 -1.57570 -1.55232 -1.50652 -6.02925
(b)
2.00000 -1.60254 -1.55903 -1.55232 -0.43239 -3.23517
9-113
(a)
2.00000
(b)
2.00000
9-115 9-117 9-119
2.47969
2.64773
9-125 After one step: 9-127 After one step: 9-129 After one step: 9-131 After one step: 9-133 After one step:
2.68943
2.81201
2.77967
2.81201
4.35908
11.81790
1.15010
5.84214
,
,
After two steps:
,
After two steps:
,
,
,
,
After two steps:
After two steps:
,
,
9-135
After 10 steps with h=0.2:
,
After 20 steps with h=0.1:
9-137
.
After 10 steps with h=0.2:
,
After 20 steps with h=0.1:
,
9-139
.
,
.
After 10 steps with h=0.2:
,
After 10 steps with h=0.2:
,
9-141
After 10 steps with h=0.2:
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.
After 20 steps with h=0.1:
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.
After 10 steps with h=0.2:
,
.
After 20 steps with h=0.1:
,
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9-143
9-145
After 10 steps with h=0.2: After 20 steps with h=0.1: 9-147
. .
After 10 steps with h=0.2:
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.
After 20 steps with h=0.1:
,
.
9-153 After one step: 9-155 After one step: 9-157 After one step:
9-159
, ,
, 0.07 After two steps:
,
After two steps:
,
, 1.89
, 8.75
After two steps:
After 10 steps with h=0.2:
,
After 20 steps with h=0.1:
9-161
After 20 steps with h=0.1:
,
.
After 10 steps with h=0.2:
,
After 20 steps with h=0.1:
,
9-173 After one step: 9-175 After one step: 9-177 After one step: 9-179 After one step:
9-183
.
,
9-171 After one step:
9-181
,
After 10 steps with h=0.2:
9-163
.
.
,
After two steps:
,
After two steps:
,
,
After two steps:
,
After two steps:
,
,
After 20 steps with h=0.1:
,
After 10 steps with h=0.2: After 20 steps with h=0.1: 9-187 After 10 steps with h=0.2: After 20 steps with h=0.1:
,
.
After 10 steps with h=0.2:
9-185
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After two steps:
After 10 steps with h=0.2:
After 20 steps with h=0.1:
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.
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, 0.00%
.
9-189
After 10 steps with h=0.2:
,
.
After 20 steps with h=0.1: 9-191
After 10 steps with h=0.2:
.
After 20 steps with h=0.1: 9-193
.
After 10 steps with h=0.2:
,
.
After 20 steps with h=0.1:
,
.
9-205 After one step: 9-207 After one step: 9-209 After one step: 9-211 After one step: 9-213 After one step: 9-215
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. After two steps:
. After two steps:
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. After two steps:
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After 20 steps with h=0.1:
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After 10 steps with h=0.2:
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After 20 steps with h=0.1:
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After 10 steps with h=0.2:
,
.
After 20 steps with h=0.1:
,
.
After 10 steps with h=0.2:
,
.
After 20 steps with h=0.1:
,
.
,
.
9-218
9-219
9-221
After 10 steps with h=0.2:
.
,
After 10 steps with h=0.2:
9-217
,
.
,
.
.
.
.
