Section 10.1
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 10.1
1.
1/s2, s>0 2.
2/s3, s>0 3.
e/(s−3), s>3 4.
s/(s2+1), s>0 5.
1/(s2−1), s>1 6.
12[1/s−s/(s2+4)], s>0 7.
(1−e−s)/s, s>0 8.
(e−s−e−2s)/s, s>0 9.
(1−e−s−se−s)/s2, s>0 10.
(s−1+e−s)/s2, s>0 11.
12π−−√s−3/2+3s−2, s>0 12.
(45π−192s3/2)/(8s7/2), s>0 13.
s−2−2(s−3)−1, s>3 14.
3π−−√/(4s5/2)+1/(s+10), s>0 15.
s−1+s(s2−25)−1, s>5 16.
(s+2)/(s2+4), s>0 17.
cos2 2t=12(1+ cos 4t) ;12[s−1+s/(s2+16)], s>0 18.
3/(s2+36), s>0 19.
s−1+3s−2+6s−3+6s−4, s>0 20.
1/(s−1)2, s>1 21.
(s2−4)/(s2+4)2, s>0 22.
12[s/(s2−36)−s−1] 23.
12t3 24.
2t/π−−−√ 25.
1−83t3/2π−1/2 26.
e−5t 27.
3e4t 28.
3 cos 2t+12sin 2t 29.
53sin 3t−3 cos 3t 30.
−cosh 2t−92 sinh 2t 31.
35 sinh 5t−10 cosh 5t 32.
2u(t−3) 37.
f(t)=1−u(t−a) . Your figure should indicate that the graph of f contains the point (a, 0), but not the point (a, 1).38.
f(t)=u(t−a)−u(t−b) . Your figure should indicate that the graph of f contains the points (a, 1) and (b, 0), but not the points (a, 0) and (b, 1).39. Figure 10.2.8 shows the graph of the unit staircase function.
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