E
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
E
Earth-Moon satellite orbits, 461
Echelon form, 150
Echelon matrix, 150
reduced, 157
Eigenspace, 345
Eigenvalue, 339
Eigenvalue matrix, 348
Eigenvalue solution, 386
Eigenvalue-eigenvector algorithm, 341
Eigenvector equation, 386
Eigenvector matrix, 348
Element (of a matrix), 147
Elementary matrix, 180
and row operations, 180
Elimination constant, 37
Elimination method:
Elliptic integral, 548
Elliptical orbit, 412
Engineering functions, 603
Entry (of a matrix), 147
Equilibrium position, 302
Equilibrium solution, 88
of system, 504
Equivalent systems and matrices, 150
Error:
Error function, 53
Escape velocity, 100
Euclidean space, inner product, 251
Euler, Leonhard (1707–1783),
Euler (differential) equation, 277
Euler's formula, 296
Euler's method, 107
Euler's theorem, 565
Exact equation, 65
Exactness, criterion for, 66
Existence, uniqueness of solutions:
Exponential growth, see Natural growth
Exponential matrix, 472
computation of, 478
Exponential order, 563
Exponential series, 606
Exponents (at a regular singular point), 632
External force, 303
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