C
by C. Henry Edwards,
David E. Penney,
David T. Calvis,
Differential Equations and Linear Algebra, 4th Edition
- Differential Equations & Linear Algebra
- Contents
- Application Modules
- Preface
- 1 First-Order Differential Equations
- 2 Mathematical Models and Numerical Methods
- 2.1 Population Models
- 2.2 Equilibrium Solutions and Stability
- 2.3 Acceleration–Velocity Models Acceleration–Velocity Models
- 2.4 Numerical Approximation: Euler’s Method Numerical Approximation: Euler’s Method
- 2.5 A Closer Look at the Euler Method
- An Improvement in Euler’s Method
- 2.6 The Runge–Kutta Method The Runge–Kutta Method
- 3 Linear Systems and Matrices
- 4 Vector Spaces
- 5 Higher-Order Linear Differential Equations
- 5.1 Introduction: Second-Order Linear Equations
- 5.2 General Solutions of Linear Equations
- 5.3 Homogeneous Equations with Constant Coefficients
- 5.4 Mechanical Vibrations
- 5.5 Nonhomogeneous Equations and Undetermined Coefficients
- 5.6 Forced Oscillations and Resonance
- 6 Eigenvalues and Eigenvectors
- 7 Linear Systems of Differential Equations
- 7.1 First-Order Systems and Applications
- 7.2 Matrices and Linear Systems
- 7.3 The Eigenvalue Method for Linear Systems
- 7.4 A Gallery of Solution Curves of Linear Systems
- Systems of Dimension n=2
- Real Eigenvalues
- Saddle Points
- Nodes: Sinks and Sources
- Zero Eigenvalues and Straight-Line Solutions
- Repeated Eigenvalues; Proper and Improper Nodes
- The Special Case of a Repeated Zero Eigenvalue
- Complex Conjugate Eigenvalues and Eigenvectors
- Pure Imaginary Eigenvalues: Centers and Elliptical Orbits
- Complex Eigenvalues: Spiral Sinks and Sources
- A 3-Dimensional Example
- 7.4 Problems
- 7.4 Application Dynamic Phase Plane Graphics
- 7.5 Second-Order Systems and Mechanical Applications*
- 7.6 Multiple Eigenvalue Solutions
- 7.7 Numerical Methods for Systems
- 7.7 Application Comets and Spacecraft
- 8 Matrix Exponential Methods
- 9 Nonlinear Systems and Phenomena
- 9.1 Stability and the Phase Plane
- 9.2 Linear and Almost Linear Systems
- 9.3 Ecological Models: Predators and Competitors
- 9.4 Nonlinear Mechanical Systems
- 10 Laplace Transform Methods
- 11 Power Series Methods
- References for Further Study
- APPENDIX A Existence and Uniqueness of Solutions
- APPENDIX B Theory of Determinants
- Answers to Selected Problems
- Index
C
Carbon-14, 36
Cart with flywheel, 327
Catenary problem, 44
Cauchy-Schwarz inequality, 251
Center:
Central conic, 208
Characteristic value, see Eigenvalue
Clairaut equation, 71
Clarinet reed, 551
Clarke, Arthur (1917–2008), 17
Closed trajectory, 509
Clepsydra, 43
Coddington, E. A. (1920–1991), 633
Coefficient matrix, 147
Coexistence of species, 532
Cofactor, 190
Column (of a matrix), 147
Column rank, 244
and row rank, 247
Column space, 244
basis for, 246
Comet orbit, 467
Compartmental analysis, 388
Competition situation, 80
Complementary function, 287
Complete eigenvalue, 437
Complete independence of eigenvectors, 353
Complete set of eigenvectors, 437
Complex-valued function, real and imaginary parts, 296
Compound interest, 36
Computational efficiency, 200
Constant acceleration, 12
Continuous dependence of solutions, 665
Convergence:
Convolution (of functions), 587
Convolution property (of transforms), 588
Cooperation of species, 534
Coordinate plane, 214
space, 211
Corresponding eigenvalue, 339
Criterion for diagonalizability, 349
Criterion for exactness, 66
Critical point (of system), 503
Cumulative error, 110
Curvature, 71
Curve fitting, 203
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