Section 3.5
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 3.5
1.
A−1=[3−4−23], x=[3−2] 2.
A−1=[5−2−73], x=[−2611] 3.
A−1=[6−5−76], x=[33−28] 4.
A−1=[1717−125], x=[25−10] 5.
A−1=12[4−5−23], x=12[8−7] 6.
A−1=13[6−3−74], x=13[25−10] 7.
A−1=14[7−5−97], x=14[3−1] 8.
A−1=15[10−5−158], x=15[25−11] 9.
A−1=[5−4−65] 10.
A−1=12[6−4−75] 11.
A−1=⎡⎣⎢−52−4−21−35−25⎤⎦⎥ 12.
A−1=⎡⎣⎢18−3−420−1−712⎤⎦⎥ 13.
A−1=⎡⎣⎢−133242−9−7−511⎤⎦⎥ 14.
A−1=⎡⎣⎢11−4−2−731−932⎤⎦⎥ 15.
A−1=⎡⎣⎢−22−271023−178−3⎤⎦⎥ 16.
A−1=13⎡⎣⎢−3−1−10−333−12⎤⎦⎥ 17.
A−1=14⎡⎣⎢−2−2−2−6−2−2−3−11⎤⎦⎥ 18.
A−1=15⎡⎣⎢1−5−320−1−256⎤⎦⎥ 19.
A−1=16⎡⎣⎢−219−311−538−20⎤⎦⎥ 20.
A−1=17⎡⎣⎢3−2−11−32070⎤⎦⎥ 21.
A−1=⎡⎣⎢⎢⎢⎢0−210100−301000001⎤⎦⎥⎥⎥⎥ 22.
A−1=⎡⎣⎢⎢⎢⎢100−3−1−2131−11−502−11⎤⎦⎥⎥⎥⎥ 23.
A−1=[4−5−34], X=[7−918−23−3545] 24.
A−1=[7−8−67], X=[14−16−303546−53] 25.
A−1=⎡⎣⎢11−2−2−9214−10⎤⎦⎥, X=⎡⎣⎢7−1−2−143215−2−4⎤⎦⎥ 26.
A−1=⎡⎣⎢−166−133−1211−4−9⎤⎦⎥, X=⎡⎣⎢−218−179−366−25⎤⎦⎥ 27.
A−1=⎡⎣⎢70−2−20−16171−5⎤⎦⎥, X=⎡⎣⎢171−5−20−16241−7−13−14⎤⎦⎥ 28.
A−1=⎡⎣⎢−5−82458−231015−45⎤⎦⎥, X=⎡⎣⎢−5−82458−231015−4517−13⎤⎦⎥
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