Ordinary differential equations:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Waltham, Mass.
Blaisdell
1969
|
| Edition: | 2. ed. |
| Series: | A Blaisdell book in pure and applied mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 366 S. Ill. u. graph. Darst. |
Staff View
MARC
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| 245 | 1 | 0 | |a Ordinary differential equations |c Garrett Birkhoff, Gian-Carlo Rota |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Waltham, Mass. |b Blaisdell |c 1969 | |
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| 490 | 0 | |a A Blaisdell book in pure and applied mathematics | |
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| 650 | 4 | |a Differential equations | |
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Record in the Search Index
| _version_ | 1804115381163393024 |
|---|---|
| adam_text | Contents
I
FIRST ORDER DIFFERENTIAL EQUATIONS
1 Introduction 1
2 Fundamental theorem of the calculus 3
3 Solutions and integrals 4
4 Regular and normal curve families 6
5 Exact differentials; integrating factors 9
6 First order linear equations 11
7 The linear fractional equation 13
8 Graphical integration 15
*9 Equations of higher degree 17
10 Initial value problems 20
11 Uniqueness and continuity 21
12 The comparison theorem 24
2
SECOND ORDER LINEAR EQUATIONS
1 Basic definitions 28
2 Initial value problems 31
3 The Wronskian 33
4 Separation and comparison theorems 37
*5 Poincare phase plane 39
6 Adjoint operators 41
7 Lagrange identity 42
8 Green s functions 44
9 Variation of parameters 47
*10 Two endpoint problems 49
*11 Green s functions 51
ix
x Contents
3
POWER SERIES SOLUTIONS
1 Introduction 56
2 Method of undetermined coefficients 58
*3 Sine and cosine functions 61
*4 Bessel functions 63
5 Analytic functions 67
6 Method of majorants 69
7 First order nonlinear differential equations 72
8 Undetermined coefficients 73
*9 Radius of convergence 76
*10 Method of majorants, II 77
*11 Complex solutions 80
4
LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS
1 The characteristic polynomial 83
2 Real and complex solutions 85
3 Linearly independent solutions 87
4 Solution bases 89
5 Stability 92
6 Inhomogeneous equations 94
7 The transfer function 96
*8 The Nyquist diagram 98
9 The Green s function 101
*10 Laplace transforms 105
*11 Convolution 109
5
PLANE AUTONOMOUS SYSTEMS
1 Autonomous systems 112
2 Plane autonomous systems 115
3 Poincare phase plane 117
4 Linear autonomous systems 120
Contents xi
5 Equivalent systems 122
6 Linear equivalence 124
7 Stability 128
8 Focal, nodal, and saddle points 131
9 Method of Liapunov 134
10 Undamped nonlinear oscillations 136
11 Soft and hard springs 138
12 Damped nonlinear oscillations 141
*13 Limit cycles 142
6
EXISTENCE AND UNIQUENESS THEOREMS
1 Introduction 148
2 Lipschitz condition 149
3 Well set problems 152
4 Continuity 154
*5 Normal systems 157
6 Equivalent integral equation 161
7 Successive approximation 162
8 Linear systems 165
9 Local existence theorem 167
*10 Analytic equations 169
11 Continuation of solutions 172
*12 The perturbation equation 174
13 Plane autonomous systems 176
*14 The Peano existence theorem 177
7
APPROXIMATE SOLUTIONS
1 Introduction 182
2 Cauchy polygons 183
3 Error bounds 187
4 Order of accuracy 188
5 Midpoint quadrature 189
6 Trapezoidal quadrature 193
*7 ^Iccwrate error estimates 194
8 Trapezoidal integration 197
xii Contents
9 Order of accuracy 198
10 The improved Euler method 200
11 The modified Euler method 202
*12 The Cumulative error 204
8
EFFICIENT NUMERICAL INTEGRATION
1 Introduction 208
2 Difference operators 209
3 Characteristic equation; stability 210
4 Polynomial interpolation 213
5 The interpolation error 215
6 Roundoff errors 217
7 Numerical differentiation 218
8 Difference approximations 221
9 Simpson s rule 223
*10 Gaussian quadrature 226
11 Milne s method 228
12 Stability; error analysis 231
13 Local power series 234
14 The Runge Kutta method 235
9
REGULAR SINGULAR POINTS
1 Continuation principle 240
*2 Movable singular points 243
3 First order equations 244
4 Circuit matrix 247
5 Canonical bases 249
6 Regular singular points 252
7 Bessel equation 254
8 The fundamental theorem 259
*9 Alternative proof of the fundamental theorem 262
*10 Hypergeometric functions 265
*11 Jacobi polynomials 266
*12 Singular points at infinity 269
*13 Fuchsian equations 271
Contents xiii
10
STURM LIOUVILLE SYSTEMS
1 Sturm Liouville systems 277
2 Sturm Liouville series 279
*3 Physical interpretations 283
4 Singular systems 284
5 Pru fer substitution 288
6 The Sturm comparison theorem 289
7 The oscillation theorem 290
8 The sequence of eigenfunctions 294
9 The Liouville normal form 296
10 Modified Pru fer substitution 299
*11 The asymptotic behavior of Bessel functions 302
12 Distribution of eigenvalues 303
13 Normalized eigenfunctions 305
14 Inhomogeneous equations 308
15 Green s functions 309
*16 The Schroedinger equation 311
*17 The square well potential 313
*18 Mixed spectrum 314
II
EXPANSIONS IN EIGENFUNCTIONS
1 Fourier series 318
2 Orthogonal expansions 320
3 Mean square approximation 321
4 Completeness 323
5 Orthogonal polynomials 326
*6 Properties of orthogonal polynomials 328
*7 Chebyshev polynomials 332
8 Euclidean vector spaces 334
9 Completeness of eigenfunctions 336
*10 Hiibert space 338
*11 Proof of completeness 341
xiv Contents
APPENDIX: LINEAR SYSTEMS
1 Matrix norm 347
2 Constant coefficient systems 348
3 The matrizant 349
4 Floquet theorem; canonical bases 351
BIBLIOGRAPHY 355
INDEX 361
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| physical | XIV, 366 S. Ill. u. graph. Darst. |
| publishDate | 1969 |
| publishDateSearch | 1969 |
| publishDateSort | 1969 |
| publisher | Blaisdell |
| record_format | marc |
| series2 | A Blaisdell book in pure and applied mathematics |
| spelling | Birkhoff, Garrett 1911-1996 Verfasser (DE-588)11882564X aut Ordinary differential equations Garrett Birkhoff, Gian-Carlo Rota 2. ed. Waltham, Mass. Blaisdell 1969 XIV, 366 S. Ill. u. graph. Darst. txt rdacontent n rdamedia nc rdacarrier A Blaisdell book in pure and applied mathematics Équations différentielles Differential equations Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 s DE-604 Differentialgleichung (DE-588)4012249-9 s 1\p DE-604 Rota, Gian-Carlo 1932-1999 Verfasser (DE-588)119286416 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000580238&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Birkhoff, Garrett 1911-1996 Rota, Gian-Carlo 1932-1999 Ordinary differential equations Équations différentielles Differential equations Differentialgleichung (DE-588)4012249-9 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| subject_GND | (DE-588)4012249-9 (DE-588)4020929-5 |
| title | Ordinary differential equations |
| title_auth | Ordinary differential equations |
| title_exact_search | Ordinary differential equations |
| title_full | Ordinary differential equations Garrett Birkhoff, Gian-Carlo Rota |
| title_fullStr | Ordinary differential equations Garrett Birkhoff, Gian-Carlo Rota |
| title_full_unstemmed | Ordinary differential equations Garrett Birkhoff, Gian-Carlo Rota |
| title_short | Ordinary differential equations |
| title_sort | ordinary differential equations |
| topic | Équations différentielles Differential equations Differentialgleichung (DE-588)4012249-9 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| topic_facet | Équations différentielles Differential equations Differentialgleichung Gewöhnliche Differentialgleichung |
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