Section 10.2
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 10.2
1.
x(t)=5 cos 2t 2.
x(t)=3 cos 3t+43sin 3t 3.
x(t)=23(e2t−e−t) 4.
x(t)=12(7e−3t−3e−5t) 5.
x(t)=13(2 sin t−sin 2t) 6.
x(t)=13(cos t− cos 2t) 7.
x(t)=18(9 cos t−cos 3t) 8.
x(t)=19(1− cos 3t) 9.
x(t)=16(2−3e−t+e−3t) 10.
x(t)=14(2t−3+12e−t−9e−2t) 11.
x(t)=1, y(t)=−2 12.
x(t)=29(e2t−e−t−3te−t), y(t)=19(e2t−e−t+6te−t) 13.
x(t)=−(2/3–√) sinh (t/3–√), y(t)=cosh (t/3–√)+(1/3–√) sinh (t/3–√) 14.
x(t)=14(2t−3 sin 2t), y(t)=−18(2t+3 sin 2t) 15.
x(t)=13(2+e−3t/2 [cos (rt/2)+r sin (rt/2)]), y(t)=121(28−9et+2e−3t/2 [cos (rt/2)+4r sin (rt/2)]) wherer=3–√ 16.
x(t)=cos t+ sin t, y(t)=et− cos t, z(t)=−2 sin t 17.
f(t)=13(e3t−1) 18.
f(t)=35(1−e−5t) 19.
f(t)=14(1− cos 2t)=12sin2 t 20.
f(t)=19(6 sin 3t− cos 3t+1) 21.
f(t)=t− sin t 22.
f(t)=19(−1+ cosh 3t) 23.
f(t)=−t+ sinh t 24.
f(t)=12(e−2t−2e−t+1)
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