A First Course in Differential Equations

400 Pages·2011·4.282 MB·English

by  J. David Logan (auth.)

Undergraduate Texts in Mathematics
Editorial Board
S. Axler
K.A. Ribet
For other titles published in this series, go to
http://www.springer.com/series/666

J. David Logan
A First Course
in Differential Equations
Second Edition
J. David Logan
Department of Mathematics
University of Nebraska—Lincoln
Lincoln, NE 68588-0130
USA
[email protected]
Editorial Board
S.Axler K.A.Ribet
Mathematics Department Mathematics Department
San Francisco State University University of California at
Berkeley Berkeley, CA 94720-3840
San Francisco, CA 94132 USA
USA [email protected]
[email protected]
ISSN 0172-6056
ISBN 978-1-4419-7591-1 e-ISBN 978-1-4419-7592-8
DOI 10.1007/978-1-4419-7592-8
Springer New York Dordrecht Heidelberg London

Library of Congress Control Number: 2010938913

Mathematics Subject Classification (2010): 34-01

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Springer is part of Springer Science+Business Media (www.springer.com)
To my son David

Contents
Preface to the Second Edition .................................. xi
To the Student ................................................. xv
1. Differential Equations and Models .......................... 1
1.1 Introduction ............................................. 1
1.2 General Terminology...................................... 10
1.2.1 Geometrical Interpretation .......................... 20
1.3 Pure Time Equations ..................................... 23
1.4 Mathematical Models ..................................... 29
1.4.1 Particle Dynamics .................................. 31
1.5 Separation of Variables.................................... 38
1.6 Autonomous Differential Equations ......................... 47
1.7 Stability and Bifurcation .................................. 59
1.8 Reactors and Circuits ..................................... 65
1.8.1 Chemical Reactors.................................. 66
1.8.2 Electrical Circuits .................................. 69
2. Linear Equations: Solutions and Approximations ........... 73
2.1 First-Order Linear Equations .............................. 73
2.2 Approximation of Solutions................................ 86
2.2.1 Picard Iteration* ................................... 86
2.2.2 Numerical Methods................................. 89
2.2.3 Error Analysis ..................................... 95
viii A First Course in Differential Equations Second Edition
3. Second-Order Differential Equations ........................103
3.1 Particle Mechanics........................................104
3.2 Linear Equations with Constant Coefficients .................111
3.3 The Nonhomogeneous Equation ............................121
3.3.1 Undetermined Coefficients...........................122
3.3.2 Resonance.........................................130
3.4 Variable Coefficients ......................................133
3.4.1 Cauchy–Euler Equation .............................135
3.4.2 Power Series Solutions* .............................137
3.4.3 Reduction of Order*................................141
3.4.4 Variation of Parameters .............................142
3.5 Steady–State Heat Conduction*............................146
3.6 Higher-Order Equations ...................................153
3.7 Summary and Review.....................................157
4. Laplace Transforms.........................................161
4.1 Definition and Basic Properties ............................161
4.2 Initial Value Problems ....................................171
4.3 The Convolution Property.................................177
4.4 Piecewise Continuous Sources ..............................181
4.5 Impulsive Sources ........................................184
4.6 Table of Laplace Transforms ...............................191
5. Systems of Differential Equations ...........................193
5.1 Linear Systems...........................................194
5.2 General Solution and Geometric Behavior ...................210
5.3 Linear Orbits ............................................217
5.4 Nonlinear Models.........................................223
5.5 Applications .............................................231
5.5.1 The Lotka–Volterra Model...........................232
5.5.2 Models in Ecology..................................236
5.5.3 An Epidemic Model ................................239
5.6 Numerical Methods.......................................245
6. Linear Systems and Matrices ...............................251
6.1 Linearization and Stability.................................251
6.2 Matrices* ...............................................254
6.3 Two-Dimensional Linear Systems...........................268
6.3.1 Matrix Formulation.................................268
6.3.2 The Eigenvalue Problem ............................274
6.3.3 Real Unequal Eigenvalues ...........................276
6.3.4 Complex Eigenvalues ...............................279
Contents ix
6.3.5 Real Repeated Eigenvalues ..........................280
6.4 Stability.................................................284
6.5 Nonhomogeneous Systems* ................................287
6.6 Three-Dimensional Systems*...............................293
7. Nonlinear Systems..........................................299
7.1 Linearization Revisited....................................299
7.1.1 Malaria* ..........................................307
7.2 Periodic Orbits...........................................316
7.3 The Poincar´e–BendixsonTheorem..........................320
Appendix A. References ........................................331
Appendix B. Computer Algebra Systems .......................333
B.1 Maple...................................................334
B.2 MATLABR ..............................................336
(cid:13)
Appendix C. Practice Test Questions ...........................341
Appendix D. Solutions and Hints to Selected Exercises .........351
E. Index.......................................................383