Section 3.6
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
Section 3.6
1. 60
2. 4
3.
−210 4.
−72 5. 120
6. 60
7. 0
8. 25
9. 30
10. 7
11. 40
12. 10
13. 78
14.
−22 15.
−74 16. 84
17. 8
18. 135
19. 39
20. 79
21.
Δ=1, x=10, y=−7 22.
Δ=1, x=−1, y=1 23.
Δ=1, x=2, y=−4 24.
Δ=1, x=5, y=−3 25.
Δ=2, x=6, y=−3 26.
Δ=−2, x=12, y=0 27.
Δ=96, x1=13, x2=−23, x3=−13 28.
Δ=35, x1=47, x2=37, x3=27 29.
Δ=23, x1=2, x2=3, x3=0 30.
Δ=56, x1=−17, x2=914, x3=27 31.
Δ=14, x1=−87, x2=−107, x3=17 32.
Δ=6, x1=−73, x2=9, x3=8 33.
det A=−4, A−1=14⎡⎣⎢416284152541323⎤⎦⎥ 34.
det A=35, A−1=14⎡⎣⎢−2913−3−4212−19−8⎤⎦⎥ 35.
det A=35, A−1=135⎡⎣⎢−15101525−5−25−26819⎤⎦⎥ 36.
det A=23, A−1=123⎡⎣⎢510120174−17−11−8⎤⎦⎥ 37.
det A=29, A−1=129⎡⎣⎢11−1718−1419−15−1510−14⎤⎦⎥ 38.
det A=6, A−1=16⎡⎣⎢−6151210−21−1826−6⎤⎦⎥ 39.
det A=37, A−1=137⎡⎣⎢−214−6−195−136−9⎤⎦⎥ 40.
det A=107, A−1=1107⎡⎣⎢911−1512−21−20−13−4−14⎤⎦⎥
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