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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
L
Lakes Erie, Huron, Ontario, 52
Laplace, Pierre Simon de (1749–1827), 565
Laplace transform, 558
and convolution, 588
and initial value problems, 568
and linear systems, 571
differentiation of, 589
existence, 563
integration of transforms, 591
inverse, 561
inverse transforms of series, 594
linearity, 560
notation, 561
of derivative, 568
of higher derivatives, 569
of integral, 574
of periodic function, 597
of square and triangular wave functions, 598
products of transforms, 587
translation on the s-axis, 579
translation on the t-axis, 594
uniqueness of inverse, 565
Law of cosines, 202
Legendre polynomial, 623
Length of vector, 213
Leonardo Fibonacci (1175–1250?), 363
Limit cycle, 524
Limited environment situation, 80
Limiting velocity, 20
Linear approximation formula, 109
Linear differential equation:
Linear factor partial fractions, 579
Linear system (algebraic), 139
Linear system (differential equations):
Linearity of Laplace transform, 560
Linearized system, 516
Lipschitz continuous, 658
Local error, 110
Local existence of solutions, 663
Logarithmic decrement, 313
Logistic populations, interactions of, 534
Logistic prey population, 536
Lower triangular matrix, 194
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