Differential equations with linear algebra:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford [u.a.]
Oxford Univ. Press
2009
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes index. |
| Physical Description: | XVII, 553 S. graph. Darst. |
| ISBN: | 9780195385861 |
Staff View
MARC
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| 100 | 1 | |a Boelkins, Matthew R. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Differential equations with linear algebra |c Matthew R. Boelkins, J. L. Goldberg and Merle C. Potter |
| 264 | 1 | |a Oxford [u.a.] |b Oxford Univ. Press |c 2009 | |
| 300 | |a XVII, 553 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes index. | ||
| 650 | 4 | |a Differential equations, Linear | |
| 650 | 4 | |a Algebras, Linear | |
| 650 | 0 | 7 | |a Lineare Algebra |0 (DE-588)4035811-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Gewöhnliche Differentialgleichung |0 (DE-588)4020929-5 |2 gnd |9 rswk-swf |
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| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Goldberg, Jack L. |e Verfasser |4 aut | |
| 700 | 1 | |a Potter, Merle C. |d 1936- |e Verfasser |0 (DE-588)138923493 |4 aut | |
| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018617029&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
Record in the Search Index
| _version_ | 1805082910219304960 |
|---|---|
| adam_text |
Contents
Introduction
1
Essentials of linear algebra
3
1.1
Motivating problems
3
1.2
Systems of linear equations
8
1.2.1
Row reduction using Maple
15
1.3
Linear combinations
21
1.3.1
Markov chains: an application of matrix-vector
multiplication
26
1.3.2
Matrix products using
Марк
29
1.4
The span of a set of vectors
33
1.5
Systems of linear equations revisited
39
1.6
Linear independence
49
1.7
Matrix algebra
58
1.7.1
Matrix algebra using Maple
62
1.8
The inverse of a matrix
66
1.8.1
Computer graphics
70
1.8.2
Matrix inverses using
Марк
73
1.9
The determinant of a matrix
78
1.9.1
Determinants using Maple
82
1.10
The eigenvalue problem
84
1.10.1
Markov chains, eigenvectors, and Google
93
1.10.2
Using
Марк
to find eigenvalues and eigenvectors
94
vi
Contents
1.11
Generalized vectors
"
1.12
Bases and dimension in vector spaces
108
1.13
For further study
! !
5
1.13.1
Computer graphics: geometry and linear algebra at
work H5
1.13.2
Bézier
curves
49
1.13.3
Discrete dynamical systems
123
2
First-order differential equations
127
2.1
Motivating problems
127
2.2
Definitions, notation, and terminology
129
2.2.1
Plotting slope fields using Maple
135
2.3
Linear first-order differential equations
139
2.4
Applications of linear first-order differential equations
147
2.4.1
Mixing problems
147
2.4.2
Exponential growth and decay
148
2.4.3
Newton's law of Cooling
150
2.5
Nonlinear first-order differential equations
154
2.5.1
Separable equations
154
2.5.2
Exact equations
157
2.6
Euler's method
162
2.6.1
Implementing Euler's method in Excel
167
2.7
Applications of nonlinear first-order differential
equations
172
2.7.1
The logistic equation
172
2.7.2
Torricelli'slaw
176
2.8
For further study
181
2.8.1
Converting certain second-order
des
to
first-order DEs
181
2.8.2
How raindrops fall
182
2.8.3
Riccati's equation
183
2.8.4
Bernoulli's equation
184
3
Linear systems of differential equations
187
3.1
Motivating problems
187
3.2
The eigenvalue problem revisited
191
3.3
Homogeneous linear first-order systems
202
3.4
Systems with all real linearly independent eigenvectors
211
3.4.1
Plotting direction fields for systems using
Марк
219
3.5
When a matrix lacks two real linearly independent
eigenvectors
223
3.6
Nonhomogeneous systems: undetermined
coefficients
236
3.7
Nonhomogeneous systems: variation of parameters
245
3.7.1
Applying variation of parameters using Maple
250
Contents
vii
3.8 Applications
of
linear
systems
253
3.8.1
Mixing problems
253
3.8.2
Spring-mass systems
255
3.8.3
RLC circuits
258
3.9
For further study
268
3.9.1
Diagonalizable matrices and coupled systems
268
3.9.2
Matrix exponential
270
Higher order differential equations
273
4.1
Motivating equations
273
4.2
Homogeneous equations: distinct real roots
274
4.3
Homogeneous equations: repeated and complex roots
281
4.3.1
Repeated roots
281
4.3.2
Complex roots
283
4.4
Nonhomogeneous equations
288
4.4.1
Undetermined coefficients
289
4.4.2
Variation of parameters
295
4.5
Forced motion: beats and resonance
300
4.6
Higher order linear differential equations
309
4.6.1
Solving characteristic equations using
Марк
316
4.7
For further study
319
4.7.1
Damped motion
319
4.7.2
Forced oscillations with damping
321
4.7.3
The Cauchy-Euler equation
323
4.7.4
Companion systems and companion matrices
325
Laplace transforms
329
5.1
Motivating problems
329
5.2
Laplace transforms: getting started
331
5.3
General properties of the Laplace transform
337
5.4
Piecewise continuous functions
347
5.4.1
The Heaviside function
347
5.4.2
The Dirac delta function
353
5.4.3
The Heaviside and Dirac functions in Maple
357
5.5
Solving rVPs with the Laplace transform
359
5.6
More on the inverse Laplace transform
371
5.6.1
Laplace transforms and inverse transforms
using Maple
375
5.7
For further study
378
5.7.1
Laplace transforms of infinite series
378
5.7.2
Laplace transforms of periodic forcing functions
380
5.7.3
Laplace transforms of systems
384
Nonlinear systems of differential equations
387
6.1
Motivating problems
387
viii Contents
6.2
Graphical behavior of solutions for
2
χ
2
nonlinear
systems 391
6.2.1
Plotting direction fields of nonlinear systems
using Maple
397
6.3
Linear approximations of nonlinear systems
400
6.4
Euler's method for nonlinear systems
409
6.4.1
Implementing Euler's method for systems in Excel
413
6.5
For further study
417
6.5.1
The damped pendulum
417
6.5.2
Competitive species
418
Numerical methods for differential equations
421
7.1
Motivating problems
421
7.2
Beyond Euler's method
423
7.2.1
Heun's method
424
7.2.2
Modified Euler's method
427
7.3
Higher order methods
430
7.3.1
Taylor methods
431
7.3.2
Runge-Kutta methods
434
7.4
Methods for systems and higher order equations
439
7.4.1
Euler's method for systems
440
7.4.2
Heun's method for systems
442
7.4.3
Runge-Kutta method for systems
443
7.4.4
Methods for higher order IVPs
445
7.5
For further study
449
7.5.1
Predator-Prey equations
449
7.5.2
Competitive species
450
7.5.3
The damped pendulum
450
Series solutions for differential equations
453
8.1
Motivating problems
453
8.2
A review of Taylor and power series
455
8.3
Power series solutions of linear equations
463
8.4
Legendre's equation
471
8.5
Three important examples
477
8.5.1
The Hermite equation
477
8.5.2
The Laguerre equation
480
8.5.3
The Bessel equation
482
8.6
The method of Frobenius
485
8.7
For further study
491
8.7.1
Taylor series for first-order differential equations
491
8.7.2
The Gamma function
491
Contents
ix
Appendix
A
Review
of integration techniques
493
Appendix
В
Complex numbers
503
Appendix
С
Roots of polynomials
509
Appendix
D
Linear transformations
513
Appendix
E
Solutions to selected exercises
523
Index
549 |
| any_adam_object | 1 |
| author | Boelkins, Matthew R. Goldberg, Jack L. Potter, Merle C. 1936- |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| spelling | Boelkins, Matthew R. Verfasser aut Differential equations with linear algebra Matthew R. Boelkins, J. L. Goldberg and Merle C. Potter Oxford [u.a.] Oxford Univ. Press 2009 XVII, 553 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Includes index. Differential equations, Linear Algebras, Linear Lineare Algebra (DE-588)4035811-2 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Gewöhnliche Differentialgleichung (DE-588)4020929-5 s Lineare Algebra (DE-588)4035811-2 s DE-604 Goldberg, Jack L. Verfasser aut Potter, Merle C. 1936- Verfasser (DE-588)138923493 aut Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018617029&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Boelkins, Matthew R. Goldberg, Jack L. Potter, Merle C. 1936- Differential equations with linear algebra Differential equations, Linear Algebras, Linear Lineare Algebra (DE-588)4035811-2 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| subject_GND | (DE-588)4035811-2 (DE-588)4020929-5 (DE-588)4123623-3 |
| title | Differential equations with linear algebra |
| title_auth | Differential equations with linear algebra |
| title_exact_search | Differential equations with linear algebra |
| title_full | Differential equations with linear algebra Matthew R. Boelkins, J. L. Goldberg and Merle C. Potter |
| title_fullStr | Differential equations with linear algebra Matthew R. Boelkins, J. L. Goldberg and Merle C. Potter |
| title_full_unstemmed | Differential equations with linear algebra Matthew R. Boelkins, J. L. Goldberg and Merle C. Potter |
| title_short | Differential equations with linear algebra |
| title_sort | differential equations with linear algebra |
| topic | Differential equations, Linear Algebras, Linear Lineare Algebra (DE-588)4035811-2 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| topic_facet | Differential equations, Linear Algebras, Linear Lineare Algebra Gewöhnliche Differentialgleichung Lehrbuch |
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