A First Course in Differential Equations with Modeling Applications

• Tongue: English
• Leaves: 464
• Edition: 11
• Category: Library

✦ Table of Contents

Cover
Title
Copyright
Preface
Contents
1. Introduction to Differential Equations
1.1 Denitions and Terminology
1.2 Initial-Value Problems
1.3 Differential Equations as Mathematical Models
Chapter 1 In Review
2. First-Order Differential Equations
2.1 Solution Curves Without a Solution
2.1.1 Direction Fields
2.1.2 Autonomous First-Order DEs
2.2 Separable Equations
2.3 Linear Equations
2.4 Exact Equations
2.5 Solutions by Substitutions
2.6 A Numerical Method
Chapter 2 In Review
3. Modeling with First-Order Differential Equations
3.1 Linear Models
3.2 Nonlinear Models
3.3 Modeling with Systems of First-Order DEs
Chapter 3 In Review
4. Higher-Order Differential Equations
4.1 Preliminary Theory—Linear Equations
4.1.1 Initial-Value and Boundary-Value Problems
4.1.2 Homogeneous Equations
4.1.3 Nonhomogeneous Equations
4.2 Reduction of Order
4.3 Homogeneous Linear Equations with Constant
Coefcients
4.4 Undetermined Coefcients—Superposition Approach
4.5 Undetermined Coefcients—Annihilator Approach
4.6 Variation of Parameters
4.7 Cauchy-Euler Equations
4.8 Green’s Functions
4.8.1 Initial-Value Problems
4.8.2 Boundary-Value Problems
4.9 Solving Systems of Linear DEs byElimination
4.10 Nonlinear Differential Equations
Chapter 4 In Review
5. Modeling with Higher-Order Differential Equations
5.1 Linear Models: Initial-Value Problems
5.1.1 Spring/Mass Systems: Free Undamped Motion
5.1.2 Spring/Mass Systems: Free Damped Motion
5.1.3 Spring/Mass Systems: Driven Motion
5.1.4 Series Circuit Analogue
5.2 Linear Models: Boundary-Value Problems
5.3 Nonlinear Models
Chapter 5 In Review
6. Series Solutions of Linear Equations
6.1 Review of Power Series
6.2 Solutions About Ordinary Points
6.3 Solutions About Singular Points
6.4 Special Functions
Chapter 6 In Review
7. The Laplace Transform
7.1 Denition of the Laplace Transform
7.2 Inverse Transforms and Transforms ofDerivatives
7.2.1 Inverse Transforms
7.2.2 Transforms of Derivatives
7.3 Operational Properties I
7.3.1 Translation on the S-Axis
7.3.2 Translation on the t-Axis
7.4 Operational Properties II
7.4.1 Derivatives of a Transform
7.4.2 Transforms of Integrals
7.4.3 Transform of a Periodic Function
7.5 The Dirac Delta Function
7.6 Systems of Linear Differential Equations
Chapter 7 In Review
8. Systems of Linear First-Order Differential Equations
8.1 Preliminary Theory—Linear Systems
8.2 Homogeneous Linear Systems
8.2.1 Distinct Real Eigenvalues
8.2.2 Repeated Eigenvalues
8.2.3 Complex Eigenvalues
8.3 Nonhomogeneous Linear Systems
8.3.1 Undetermined Coefcients
8.3.2 Variation of Parameters
8.4 Matrix Exponential
Chapter 8 In Review
9. Numerical Solutions of Ordinary Differential Equations
9.1 Euler Methods and Error Analysis
9.2 Runge-Kutta Methods
9.3 Multistep Methods
9.4 Higher-Order Equations and Systems
9.5 Second-Order Boundary-Value Problems
Chapter 9 In Review
APPENDIXES
A: Integral-Defined Functions
B: Matrices
C: Laplace Transforms
ANSWERS for Selected Odd-Numbered Problems
1.
1.1
1.2
1.3
R
2.
2.1
2.2
2.3
2.4
2.5
2.6
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3.
3.1
3.2
3.3
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4.
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
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5.
5.1
5.2
5.3
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6.
6.1
6.2
6.3
6.4
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7.
7.1
7.2
7.3
7.4
7.5
7.6
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8.
8.1
8.2
8.3
8.4
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9.
9.1
9.2
9.3
9.4
9.5
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appA
appB
Index
A
B
C
D
E
F
G
H
I
K
L
M
N
O
P
Q
R
S
T
U
V
W
Y
Z