Section 3.2
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 3.2
1.
x1=1, x2=0, x3=2 2.
x1=5, x2=1, x3=−3 3.
x1=13+11t, x2=2+5t, x3=t 4.
x1=35+33t, x2=5+7t, x3=t 5.
x1=13+4t, x2=6+t, x3=5+3t, x4=t 6.
x1=17+t, x2=11+3t, x3=t, x4=−4 7.
x1=3−8s+19t, x2=7+2s−7t, x3=s, x4=t 8.
x1=−25+10s+22t, x2=s, x3=10−3t, x4=t 9.
x1=1, x2=3, x3=−5, x4=6 10.
x1=63s−16t, x2=13s−8t, x3=s, x4=5t, x5=t 11.
x1=3, x2=−2, x3=4 12.
x1=5, x2=−3, x3=2 13.
x1=4+3t, x2=3−2t, x3=t 14.
x1=5+2t, x2=t, x3=7 15. Inconsistent—no solution
16. Inconsistent—no solution
17.
x1=3−2t, x2=−4+t, x3=5−3t, x4=t 18.
x1=4+2s−3t, x2=s, x3=3−4t, x4=t 19.
x1=3−s−t, x2=5+2s−3t, x3=s, x4=t 20.
x1=2+3t, x2=1+s−2t, x3=2+2s, x4=s, x5=t 21.
x1=2, x2=1, x3=3, x4=4 22.
x1=3, x2=−2, x3=4, x4=−1 23. (a) None (b)
k≠2 (c)k=2 24. (a)
k≠2 (b) None (c)k=4 25. (a)
k≠4 (b)k=4 (c) None26. (a) All k (b) None (c) None
27. (a) None (b)
k≠11 (c)k=11 28. No solution unless
c=2a+3b , in which case it has infinitely many solutions.
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