An introduction to ordinary differential equations:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2007
|
| Edition: | 1. publ., repr. with corr. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 399 S. Ill., graph. Darst. |
| ISBN: | 9780521533911 |
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MARC
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| 100 | 1 | |a Robinson, James C. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a An introduction to ordinary differential equations |c James C. Robinson |
| 250 | |a 1. publ., repr. with corr. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2007 | |
| 300 | |a XIV, 399 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 7 | |a Equações diferenciais ordinárias |2 larpcal | |
| 650 | 4 | |a Differential equations | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-016072260 | ||
Record in the Search Index
| _version_ | 1804137123945644032 |
|---|---|
| adam_text | Contents
Preface page
xiii
Introduction
1
Part I First order differential equations
3
1
Radioactive decay and carbon dating
5
1.1
Radioactive decay
5
1.2
Radiocarbon dating
6
Exercises
8
2
Integration variables
9
3
Classification of differential equations
11
3.1
Ordinary and partial differential equations
11
3.2
The order of a differential equation
13
3.3
Linear and nonlinear
13
3.4
Different types of solution
14
Exercises
16
4
*Graphical representation of solutions
using
Matlab
18
Exercises
21
5
Trivial differential equations
22
5.1
The Fundamental Theorem of Calculus
22
5.2
General solutions and initial conditions
25
5.3
Velocity, acceleration and Newton s second law
of motion
29
5.4
An equation that we cannot solve explicitly
32
Exercises
33
VUl
Contents
Part II
6
Existence and uniqueness of solutions
38
6.1
The case for an abstract result
38
6.2
The existence and uniqueness theorem
40
6.3
Maximal interval of existence
41
6.4
The Clay Mathematics Institute s
$1 000 000
question
42
Exercises
44
7
Scalar autonomous ODEs
46
„
7.1
The qualitative approach
46
7.2
Stability, instability and bifurcation
48
7.3
Analytic conditions for stability and instability
49
7.4
Structural stability and bifurcations
50
7.5
Some examples
50
7.6
The pitchfork bifurcation
54
7.7
Dynamical systems
56
Exercises
56
8
Separable equations
59
8.1
The solution recipe
59
8.2
The linear equation
і
=
λχ
61
8.3
Malthus
population model
62
8.4
Justifying the method
64
8.5
A more realistic population model
66
8.6
Further examples
68
Exercises
72
9
First order linear equations and the integrating factor
75
9.1
Constant coefficients
75
9.2
Integrating factors
76
9.3
Examples
78
9.4
Newton s law of cooling
79
Exercises
86
10
Two tricks for nonlinear equations
89
10.1
Exact equations
89
10.2
Substitution methods
94
Exercises
97
Second order linear equations with constant coefficients
99
11
Second order linear equations: general theory
101
11.1
Existence and uniqueness
101
1
1
.2
Linearity
102
11.3
Linearly independent solutions
104
11.4
*The Wronskian
106
Contents
їх
11.5
*Linear algebra
1
07
Exercises
109
12
Homogeneous second order linear equations 111
12.1
Two distinct real roots
112
12.2
A repeated real root
113
12.3
No real roots
115
Exercises
118
13
Oscillations
120
13.1
The spring
120
13.2
The simple pendulum
122
13.3
Damped oscillations
123
Exercises
126
14
Inhomogeneous second order linear equations
131
14.1
Complementary function and particular integral
131
14.2
When
ƒ (/)
is a polynomial
133
14.3
When
ƒ (0
is an exponential
135
14.4
When
ƒ (?)
is a sine or cosine
137
14.5
Rule of thumb
139
14.6
More complicated functions
ƒ
(ŕ)
139
Exercises
140
15
Resonance
141
15.1
Periodic forcing
141
15.2
Pseudo
resonance in physical systems
145
Exercises
148
16
Higher order linear equations
150
16.1
Complementary function and particular
integral
150
16.2
*The general theory for nth order equations
152
Exercises
153
Part III Linear second order equations with
variable coefficients
157
17
Reduction of order
159
Exercises
162
18
*The variation of constants formula
164
Exercises
168
19
*Cauchy-Euler equations
170
19.1
Two real roots
171
19.2
A repeated root
171
19.3
Complex roots
173
Exercises
174
χ
Contents
20
*Series solutions
of
second
order linear equations
176
20.1
Power series
176
20.2
Ordinary points
178
20.3
Regular singular points
183
20.4
Besseľs
equation
187
Exercises
195
Part IV Numerical methods and difference equations
199
21
Euler s method
201
21.1
Euler s method
201
21.2
An example
203
21.3
*Matlab implementation of Euler s method
204
21.4
Convergence of Euler s method
206
Exercises
209
22
Difference equations
213
22.1
First order difference equations
213
22.2
Second order difference equations
215
22.3
The homogeneous equation
215
22.4
Particular solutions
219
Exercises
222
23
Nonlinear first order difference equations
224
23.1
Fixed points and stability
224
23.2
Cobweb diagrams
225
23.3
Periodic orbits
226
23.4
Euler s method for autonomous equations
227
Exercises
230
24
The logistic map
233
24.1
Fixed points and their stability
234
24.2
Periodic orbits
234
24.3
The period-doubling cascade
237
24.4
The bifurcation diagram and more periodic orbits
238
24.5
Chaos
240
24.6 *
Analysis of
*„+]
=4јси(1
-
xn)
242
Exercises
245
Part V Coupled linear equations
247
25
*Vector first order equations and higher order equations
249
25.1
Existence and uniqueness for second order
equations
251
Exercises
252
26
Explicit solutions of coupled linear systems
253
Exercises
257
Contents xi
27
Eigenvalues and eigenvectors
259
27.1
Rewriting the equation in matrix form
259
27.2
Eigenvalues and eigenvectors
260
27.3
*Eigenvaiues and eigenvectors with
Matlab
266
Exercises
267
28
Distinct real eigenvalues
269
28.1
The explicit solution
270
28.2
Changing coordinates
271
28.3
Phase diagrams for uncoupled equations
276
28.4
Phase diagrams for coupled equations
279
28.5
Stable and unstable manifolds
281
Exercises
282
29
Complex eigenvalues
285
29.1
The explicit solution
285
29.2
Changing coordinates and the phase portrait
287
29.3
The phase portrait for the original equation
291
Exercises
292
30
A repeated real eigenvalue
295
30.1
A is a multiple of the identity: stars
295
30.2
A is not a multiple of the identity: improper
nodes
295
Exercises
299
31
Summary of phase portraits for linear equations
301
31.1 *
Jordan canonical form
301
Exercises
305
Part VI Coupled nonlinear equations
307
32
Coupled nonlinear equations
309
32.1
Some comments on phase portraits
309
32.2
Competition of species
310
32.3
Direction fields
311
32.4
Analytical method for phase portraits
314
Exercises
322
33
Ecological models
323
33.1
Competing species
323
33.2
Predator-prey models I
331
33.3
Predator-prey models II
334
Exercises
338
34
Newtonian dynamics
341
34.1
One-dimensional conservative systems
341
34.2
*A bead on a wire
344
xii
Contents
34.3
Dissipative systems
347
Exercises
350
35
The real pendulum
352
35.1
The undamped pendulum
352
35.2
The damped pendulum
356
35.3
Alternative phase space
358
Exercises
358
36
*Periodic orbits
360
36.1
Dulac s criterion
360
36.2
The
Poinacré-Bendixson
Theorem
361
Exercises
362
-37
*The
Lorenz
equations
364
38
What next?
373
38.1
Partial differential equations and boundary
value problems
373
38.2
Dynamical systems and chaos
374
Exercises
375
Appendix A Real and complex numbers
379
Appendix
В
Matrices, eigenvalues, and eigenvectors
382
Appendix
С
Derivatives and partial derivatives
387
Index
395
|
| adam_txt |
Contents
Preface page
xiii
Introduction
1
Part I First order differential equations
3
1
Radioactive decay and carbon dating
5
1.1
Radioactive decay
5
1.2
Radiocarbon dating
6
Exercises
8
2
Integration variables
9
3
Classification of differential equations
11
3.1
Ordinary and partial differential equations
11
3.2
The order of a differential equation
13
3.3
Linear and nonlinear
13
3.4
Different types of solution
14
Exercises
16
4
*Graphical representation of solutions
using
Matlab
18
Exercises
21
5
'Trivial' differential equations
22
5.1
The Fundamental Theorem of Calculus
22
5.2
General solutions and initial conditions
25
5.3
Velocity, acceleration and Newton's second law
of motion
29
5.4
An equation that we cannot solve explicitly
32
Exercises
33
VUl
Contents
Part II
6
Existence and uniqueness of solutions
38
6.1
The case for an abstract result
38
6.2
The existence and uniqueness theorem
40
6.3
Maximal interval of existence
41
6.4
The Clay Mathematics Institute's
$1 000 000
question
42
Exercises
44
7
Scalar autonomous ODEs
46
„
7.1
The qualitative approach
46
7.2
Stability, instability and bifurcation
48
7.3
Analytic conditions for stability and instability
49
7.4
Structural stability and bifurcations
50
7.5
Some examples
50
7.6
The pitchfork bifurcation
54
7.7
Dynamical systems
56
Exercises
56
8
Separable equations
59
8.1
The solution'recipe'
59
8.2
The linear equation
і
=
λχ
61
8.3
Malthus'
population model
62
8.4
Justifying the method
64
8.5
A more realistic population model
66
8.6
Further examples
68
Exercises
72
9
First order linear equations and the integrating factor
75
9.1
Constant coefficients
75
9.2
Integrating factors
76
9.3
Examples
78
9.4
Newton's law of cooling
79
Exercises
86
10
Two 'tricks' for nonlinear equations
89
10.1
Exact equations
89
10.2
Substitution methods
94
Exercises
97
Second order linear equations with constant coefficients
99
11
Second order linear equations: general theory
101
11.1
Existence and uniqueness
101
1
1
.2
Linearity
102
11.3
Linearly independent solutions
104
11.4
*The Wronskian
106
Contents
їх
11.5
*Linear algebra
1
07
Exercises
109
12
Homogeneous second order linear equations 111
12.1
Two distinct real roots
112
12.2
A repeated real root
113
12.3
No real roots
115
Exercises
118
13
Oscillations
120
13.1
The spring
120
13.2
The simple pendulum
122
13.3
Damped oscillations
123
Exercises
126
14
Inhomogeneous second order linear equations
131
14.1
Complementary function and particular integral
131
14.2
When
ƒ (/)
is a polynomial
133
14.3
When
ƒ (0
is an exponential
135
14.4
When
ƒ (?)
is a sine or cosine
137
14.5
Rule of thumb
139
14.6
More complicated functions
ƒ
(ŕ)
139
Exercises
140
15
Resonance
141
15.1
Periodic forcing
141
15.2
Pseudo
resonance in physical systems
145
Exercises
148
16
Higher order linear equations
150
16.1
Complementary function and particular
integral
150
16.2
*The general theory for nth order equations
152
Exercises
153
Part III Linear second order equations with
variable coefficients
157
17
Reduction of order
159
Exercises
162
18
*The variation of constants formula
164
Exercises
168
19
*Cauchy-Euler equations
170
19.1
Two real roots
171
19.2
A repeated root
171
19.3
Complex roots
173
Exercises
174
χ
Contents
20
*Series solutions
of
second
order linear equations
176
20.1
Power series
176
20.2
Ordinary points
178
20.3
Regular singular points
183
20.4
Besseľs
equation
187
Exercises
195
Part IV Numerical methods and difference equations
199
21
Euler's method
201
21.1
Euler's method
201
21.2
An example
203
21.3
*Matlab implementation of Euler's method
204
21.4
Convergence of Euler's method
206
Exercises
209
22
Difference equations
213
22.1
First order difference equations
213
22.2
Second order difference equations
215
22.3
The homogeneous equation
215
22.4
Particular solutions
219
Exercises
222
23
Nonlinear first order difference equations
224
23.1
Fixed points and stability
224
23.2
Cobweb diagrams
225
23.3
Periodic orbits
226
23.4
Euler's method for autonomous equations
227
Exercises
230
24
The logistic map
233
24.1
Fixed points and their stability
234
24.2
Periodic orbits
234
24.3
The period-doubling cascade
237
24.4
The bifurcation diagram and more periodic orbits
238
24.5
Chaos
240
24.6 *
Analysis of
*„+]
=4јси(1
-
xn)
242
Exercises
245
Part V Coupled linear equations
247
25
*Vector first order equations and higher order equations
249
25.1
Existence and uniqueness for second order
equations
251
Exercises
252
26
Explicit solutions of coupled linear systems
253
Exercises
257
Contents xi
27
Eigenvalues and eigenvectors
259
27.1
Rewriting the equation in matrix form
259
27.2
Eigenvalues and eigenvectors
260
27.3
*Eigenvaiues and eigenvectors with
Matlab
266
Exercises
267
28
Distinct real eigenvalues
269
28.1
The explicit solution
270
28.2
Changing coordinates
271
28.3
Phase diagrams for uncoupled equations
276
28.4
Phase diagrams for coupled equations
279
28.5
Stable and unstable manifolds
281
Exercises
282
29
Complex eigenvalues
285
29.1
The explicit solution
285
29.2
Changing coordinates and the phase portrait
287
29.3
The phase portrait for the original equation
291
Exercises
292
30
A repeated real eigenvalue
295
30.1
A is a multiple of the identity: stars
295
30.2
A is not a multiple of the identity: improper
nodes
295
Exercises
299
31
Summary of phase portraits for linear equations
301
31.1 *
Jordan canonical form
301
Exercises
305
Part VI Coupled nonlinear equations
307
32
Coupled nonlinear equations
309
32.1
Some comments on phase portraits
309
32.2
Competition of species
310
32.3
Direction fields
311
32.4
Analytical method for phase portraits
314
Exercises
322
33
Ecological models
323
33.1
Competing species
323
33.2
Predator-prey models I
331
33.3
Predator-prey models II
334
Exercises
338
34
Newtonian dynamics
341
34.1
One-dimensional conservative systems
341
34.2
*A bead on a wire
344
xii
Contents
34.3
Dissipative systems
347
Exercises
350
35
The 'real' pendulum
352
35.1
The undamped pendulum
352
35.2
The damped pendulum
356
35.3
Alternative phase space
358
Exercises
358
36
*Periodic orbits
360
36.1
Dulac's criterion
360
36.2
The
Poinacré-Bendixson
Theorem
361
Exercises
362
-37
*The
Lorenz
equations
364
38
What next?
373
38.1
Partial differential equations and boundary
value problems
373
38.2
Dynamical systems and chaos
374
Exercises
375
Appendix A Real and complex numbers
379
Appendix
В
Matrices, eigenvalues, and eigenvectors
382
Appendix
С
Derivatives and partial derivatives
387
Index
395 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Robinson, James C. |
| author_facet | Robinson, James C. |
| author_role | aut |
| author_sort | Robinson, James C. |
| author_variant | j c r jc jcr |
| building | Verbundindex |
| bvnumber | BV022867103 |
| classification_rvk | SK 520 |
| ctrlnum | (OCoLC)231827084 (DE-599)BVBBV022867103 |
| dewey-full | 515.352 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.352 |
| dewey-search | 515.352 |
| dewey-sort | 3515.352 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 1. publ., repr. with corr. |
| format | Book |
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| id | DE-604.BV022867103 |
| illustrated | Illustrated |
| index_date | 2024-07-02T18:45:41Z |
| indexdate | 2024-07-09T21:07:18Z |
| institution | BVB |
| isbn | 9780521533911 |
| language | English |
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| oclc_num | 231827084 |
| open_access_boolean | |
| owner | DE-384 DE-19 DE-BY-UBM |
| owner_facet | DE-384 DE-19 DE-BY-UBM |
| physical | XIV, 399 S. Ill., graph. Darst. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| spelling | Robinson, James C. Verfasser aut An introduction to ordinary differential equations James C. Robinson 1. publ., repr. with corr. Cambridge [u.a.] Cambridge Univ. Press 2007 XIV, 399 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Equações diferenciais ordinárias larpcal Differential equations Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 s DE-604 Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016072260&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Robinson, James C. An introduction to ordinary differential equations Equações diferenciais ordinárias larpcal Differential equations Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| subject_GND | (DE-588)4020929-5 |
| title | An introduction to ordinary differential equations |
| title_auth | An introduction to ordinary differential equations |
| title_exact_search | An introduction to ordinary differential equations |
| title_exact_search_txtP | An introduction to ordinary differential equations |
| title_full | An introduction to ordinary differential equations James C. Robinson |
| title_fullStr | An introduction to ordinary differential equations James C. Robinson |
| title_full_unstemmed | An introduction to ordinary differential equations James C. Robinson |
| title_short | An introduction to ordinary differential equations |
| title_sort | an introduction to ordinary differential equations |
| topic | Equações diferenciais ordinárias larpcal Differential equations Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| topic_facet | Equações diferenciais ordinárias Differential equations Gewöhnliche Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016072260&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT robinsonjamesc anintroductiontoordinarydifferentialequations |