Section 2.1
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 2.1
1.
x(t)=22−e−t 
2.
x(t)=101+9e−10t 
3.
x(t)=2+e−2t2−e−2t 
4.
x(t)=3(1−e−12t)2(1+e−12t) 
5.
x(t)=408−3e−15t 
6.
x(t)=102+3e15t 
7.
x(t)=7711−4e−28t 
8.
x(t)=22117−4e91t 
9. 484
10. 20 weeks
11. (b)
P(t)=(12t+10)2 12.
P(t)=24020−t 13.
P(t)=18030−t 14.
P(t)=P01+kP0t 16. About 27.69 months
17. About 44.22 months
19. About 24.41 months
20. About 42.12 months
21.
2001+e−6/5≈153.7 million 22. About 34.66 days
23. (a)
limt→∞ x(t)=200 grams (b)54ln 3≈1.37 seconds 24. About 9.24 days
25. (a)
M=100 andk=0.0002 ; (b) In the year 207526.
50 ln 98≈15.89 months 27. (a)
100 ln95≈58.78 months ; (b)100 ln 2≈69.31 months .28. (a) The alligators eventually die out.(b) Doomsday occurs after about 9 years 2 months.
29. (a)
P(140)≈127.008 million (b) About 210.544 million; (c) In 2000 we getP≈196.169 , whereas the actual 2000 population was about 281.422 million.31.
a≈0.3915; 2.15×106 cells 37.
k≈0.0000668717, M≈338.027 38.
k≈0.000146679, M≈208.250 39.
P(t)=P0exp(kt+b2πsin 2 πt) ; the colored curve in the figure below shows the graph withP0=100, k=0.03 , andb=0.06 . It oscillates about the black curve which represents natural growth withP0=100 andk=0.03 . We see that the two agree at the end of each full year.
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