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Coordinated Differential Equations
Thomas W. Judson, Austin Eide, Adam Larios
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Front Matter
Colophon
Dedication
Preface
Preface
Acknowledgements
1
A First Look at Differential Equations
1.1
Modeling with Differential Equations
1.1.1
Exponential Growth
1.1.2
Logistic Growth
1.1.3
A Spring-Mass Model
1.1.4
A Predator-Prey System
1.1.5
Modeling the HIV-1 Virus
1.1.6
Some Questions for Thought
1.1.7
Important Lessons
1.1.8
Reading Questions
1.1.9
Exercises
1.2
Separable Differential Equations
1.2.1
Separable Differential Equations
1.2.2
Newton’s Law of Cooling
1.2.3
Mixing Problems
1.2.4
A Retirement Model
1.2.5
Some Theory
1.2.6
What Can Go Wrong
1.2.7
Important Lessons
1.2.8
Reading Questions
1.2.9
Exercises
1.3
Geometric and Quantitative Analysis
1.3.1
RC Circuits
1.3.2
Direction Fields
1.3.3
RC Circuits Revisited
1.3.4
Autonomous Differential Equations
1.3.5
Important Lessons
1.3.6
Reading Questions
1.3.7
Exercises
1.3.8
Sage—Plotting Direction Fields and Solutions
1.3.8.1
Plotting direction fields
1.3.8.2
Plotting solutions
1.3.8.3
Sage Exercises
1.4
Analyzing Equations Numerically
1.4.1
Euler’s Method
1.4.2
Finding an Error Bound
1.4.3
Improving Euler’s Method
1.4.4
Important Lessons
1.4.5
Reading Questions
1.4.6
Exercises
1.5
First-Order Linear Equations
1.5.1
Mine Tailings
1.5.2
First-Order Linear Equations
1.5.3
Mixing Models
1.5.4
Finance Models
1.5.5
Existence and Uniqueness of Solutions
1.5.6
Important Lessons
1.5.7
Reading Questions
1.5.8
Exercises
1.6
Existence and Uniqueness of Solutions
1.6.1
The Existence and Uniqueness Theorem
1.6.2
Picard Iteration
1.6.3
Important Lessons
1.6.4
Reading Questions
1.6.5
Exercises
1.7
Bifurcations
1.7.1
The Logistic Model with Harvesting Revisited
1.7.2
One-Parameter Families
1.7.3
Important Lessons
1.7.4
Reading Questions
1.7.5
Exercises
1.8
Complex Numbers and Differential Equations
1.8.1
Complex Functions and Calculus
1.8.2
Complex Differential Equations
1.8.3
The Polar Form of Complex Numbers
1.8.4
Applications to Differential Equations
1.8.5
Reading Questions
1.8.6
Exercises
1.9
Projects for First-Order Differential Equations
2
Second-Order Linear Equations
2.1
Homogeneous Linear Equations
2.1.1
RLC Circuits
2.1.2
Solving Second-Order Linear Equations
2.1.3
General Solutions to Second-Order Linear Equations
2.1.4
Classifying Harmonic Oscillators
2.1.5
Important Lessons
2.1.6
Reading Questions
2.1.7
Exercises
2.2
Forcing
2.2.1
Nonhomogeneous Equations
2.2.2
Forcing Terms
2.2.3
The Method of Undetermined Coefficients
2.2.4
A Strategy
2.2.5
Important Lessons
2.2.6
Reading Questions
2.2.7
Exercises
2.3
Sinusoidal Forcing
2.3.1
Complexification
2.3.2
Qualitative Analysis
2.3.3
Important Lessons
2.3.4
Reading Questions
2.3.5
Exercises
2.4
Forcing and Resonance
2.4.1
Resonance
2.4.2
Beats or the Case
\(\omega \neq \omega_0\)
2.4.3
Forced Damped Harmonic Motion
2.4.4
Important Lessons
2.4.5
Reading Questions
2.4.6
Exercises
2.5
Projects for Second-Order Differential Equations
3
The Laplace Transform
3.1
The Laplace Transform
3.1.1
Definition of the Laplace Transform
3.1.2
Properties of the Laplace Transform
3.1.3
Existence and Uniqueness of the Laplace Transform
3.1.4
Finding Laplace Transforms and Inverse Transforms
3.1.5
Important Lessons
3.1.6
Reading Questions
3.1.7
Exercises
3.2
Solving Initial Value Problems
3.2.1
Laplace Transforms of the Derivative
3.2.2
Discontinuous Functions
3.2.3
Forced Harmonic Oscillators
3.2.4
Important Lessons
3.2.5
Reading Questions
3.2.6
Exercises
3.3
Delta Functions and Forcing
3.3.1
Impulse Forcing
3.3.2
The Laplace Transform of the Dirac Delta Function
3.3.3
Important Lessons
3.3.4
Reading Questions
3.3.5
Exercises
3.4
Convolution
3.4.1
Convolution
3.4.2
Applying the Convolution Theorem
3.4.3
Important Lessons
3.4.4
Reading Questions
3.4.5
Exercises
3.5
Projects for Laplace Transforms
4
Systems of Differential Equations
4.1
Modeling with Systems
4.1.1
Predator-Prey Systems
4.1.2
The Spring-Mass Model Revisited
4.1.3
Modeling Epidemics
4.1.4
Important Lessons
4.1.5
Reading Questions
4.1.6
Exercises
4.2
The Geometry of Systems
4.2.1
Direction Fields
4.2.2
Modified Predator-Prey System
4.2.3
A Competing Species Model
4.2.4
A Summary of Phase Plane Analysis
4.2.5
Important Lessons
4.2.6
Reading Questions
4.2.7
Exercises
4.3
Numerical Techniques for Systems
4.3.1
Duffing’s Equation
4.3.2
Euler’s Method for Systems
4.3.3
Taylor Series Methods
4.3.4
A Word about Existence and Uniqueness
4.3.5
Important Lessons
4.3.6
Reading Questions
4.3.7
Exercises
4.4
Solving Systems Analytically
4.4.1
Partially Coupled Systems
4.4.2
Harmonic Oscillators
4.4.3
Important Lessons
4.4.4
Reading Questions
4.4.5
Exercises
4.5
Projects for Systems of Differential Equations
5
Linear Systems and Linearization
5.1
Linear Algebra in a Nutshell
5.1.1
Matrices and Systems of Linear Equations
5.1.2
Linear Independence
5.1.3
Finding Eigenvalues and Eigenvectors
5.1.4
Important Lessons
5.1.5
Reading Questions
5.1.6
Exercises
5.2
Planar Systems
5.2.1
Planar Systems and
\(2 \times 2\)
Matrices
5.2.2
Systems of Differential Equations
5.2.3
Solving Linear Systems
5.2.4
Important Lessons
5.2.5
Reading Questions
5.2.6
Exercises
5.3
Phase Plane Analysis of Linear Systems
5.3.1
The Case
\(\lambda_1 \lt 0 \lt \lambda_2\)
5.3.2
The Case
\(\lambda_1 \lt \lambda_2 \lt 0\)
5.3.3
The Case
\(\lambda_1 \gt \lambda_2 \gt 0\)
5.3.4
Important Lessons
5.3.5
Reading Questions
5.3.6
Exercises
5.4
Complex Eigenvalues
5.4.1
Complex Eigenvalues
5.4.2
Spiral Sinks and Sources
5.4.3
Solving Systems with Complex Eigenvalues
5.4.4
Important Lessons
5.4.5
Reading Questions
5.4.6
Exercises
5.5
Repeated Eigenvalues
5.5.1
Repeated Eigenvalues
5.5.2
Solving Systems with Repeated Eigenvalues
5.5.3
Important Lessons
5.5.4
Reading Questions
5.5.5
Exercises
5.6
Linearization
5.6.1
Equilibrium Solutions
5.6.2
When Linearization Fails
5.6.3
Important Lessons
5.6.4
Reading Questions
5.6.5
Exercises
6
Additional Topics
6.1
Changing Coordinates
6.1.1
Linear Maps
6.1.2
Changing Coordinates
6.1.3
Systems and Changing Coordinates
6.1.4
Distinct Real Eigenvalues
6.1.5
Complex Eigenvalues
6.1.6
Repeated Eigenvalues
6.1.7
Important Lessons
6.1.8
Reading Questions
6.1.9
Exercises
6.2
The Trace-Determinant Plane
6.2.1
The Trace-Determinant Plane
6.2.2
Parameterized Families of Linear Systems
6.2.3
Important Lessons
6.2.4
Reading Questions
6.2.5
Exercises
6.3
Linear Systems in Higher Dimensions
6.3.1
Higher-Order Linear Systems
6.3.2
The Geometry of Solutions
6.3.3
The Double Spring-Mass Systems Revisited
6.3.4
Important Lessons
6.3.5
Reading Questions
6.3.6
Exercises
6.4
The Matrix Exponential
6.4.1
The Exponential of a Matrix
6.4.2
Generalized Eigenvalues
6.4.3
Important Lessons
6.4.4
Reading Questions
6.4.5
Exercises
6.5
Hamiltonian Systems
6.5.1
The Nonlinear Pendulum
6.5.2
Hamiltonian Systems
6.5.3
The Ideal Pendulum Revisited
6.5.4
Equilibrium Solutions of Hamiltonian Systems
6.5.5
Important Lessons
6.5.6
Reading Questions
6.5.7
Exercises
6.6
More Nonlinear Mechanics
6.6.1
The Nonlinear Pendulum and Damping
6.6.2
Lyapunov Functions
6.6.3
Gradient Systems
6.6.4
Important Lessons
6.6.5
Reading Questions
6.6.6
Exercises
6.7
The Hopf Bifurcation
6.7.1
Bifurcations
6.7.2
The Hopf Bifurcation
6.7.3
Important Lessons
6.7.4
Reading Questions
6.7.5
Exercises
6.8
Projects
Reference
A
Review for students
A.1
Brief review
A.1.1
Completing the Square
A.1.2
Solving Quadratic Equations
A.1.3
Solving
\(2 \times 2\)
linear systems
A.1.4
Basic derivatives
A.1.5
Basic integrals
A.1.6
Integration by u-substitution
A.1.7
Integration by parts
A.1.8
Integration by partial fractions
B
GNU Free Documentation License
B.1
0. PREAMBLE
B.2
1. APPLICABILITY AND DEFINITIONS
B.3
2. VERBATIM COPYING
B.4
3. COPYING IN QUANTITY
B.5
4. MODIFICATIONS
B.6
5. COMBINING DOCUMENTS
B.7
6. COLLECTIONS OF DOCUMENTS
B.8
7. AGGREGATION WITH INDEPENDENT WORKS
B.9
8. TRANSLATION
B.10
9. TERMINATION
B.11
10. FUTURE REVISIONS OF THIS LICENSE
B.12
11. RELICENSING
B.13
ADDENDUM: How to use this License for your documents
C
Hints and Answers to Selected Exercises
D
Notation
Readings and References
Index
Colophon
Appendix
D
Notation
The following table defines the notation used in this book. Page numbers or references refer to the first appearance of each symbol.
Symbol
Description
Location
