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Interactive Differential Equations:
A Step-by-Step Approach to Methods & Modeling
Geoffrey W. Cox, Ph.D., ?
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Front Matter
Colophon
Preface
I
Fundamentals
1
What’s a Differential Equation?
1.1
An Analogy
1.1
Check-Point Questions
1.2
Definition
1.2
Check-Point Questions
1.3
Variables
1.3
Check-Point Questions
1.4
Terms & Coefficients
1.4
Check-Point Questions
1.5
Order
1.5
Check-Point Questions
1.6
Linear Terms
1.6
Check-Point Questions
1.7
Linearity
1.7
Check-Point Questions
1.8
Summary & Exercises
1.8
Exercises
1.8
Exercises
2
Solutions
2.1
What is a Solution?
2.1
Check-Point Questions
2.2
Verifying Solutions
2.2
Check-Point Questions
2.3
Types of Solutions
2.3
Check-Point Questions
2.4
Visualizing Solutions
2.4
Check-Point Questions
2.5
Initial Conditions & Particular Solutions
2.5
Check-Point Questions
2.6
Summary & Exercises
2.6
Exercises
II
First-Order Methods
3
Direct Integration
3.1
Antiderivatives
3.1
Check-Point Questions
3.2
Solutions by Direct Integration
3.2
Check-Point Questions
3.3
Summary & Exercises
3.3
Exercises
4
Separation of Variables
4.1
Separable Form
4.1
Check-Point Questions
4.2
Verifying Separable
4.2
Check-Point Questions
4.3
Separation of Variables Method (SOV)
4.3
Check-Point Questions
4.4
Additional Examples
4.4
Check-Point Questions
4.5
Summary & Exercises
4.5
Exercises
5
Integrating Factor
5.1
Product Rule
5.1
Check-Point Questions
5.2
The Integrating Factor
5.2
Check-Point Questions
5.3
Integrating Factor Method
5.3
Check-Point Questions
5.4
Additional Examples
5.4
Check-Point Questions
5.5
Summary & Exercises
5.5
Exercises
III
Linear Constant Coefficient Methods
6
Homogeneous
6.1
Classification
6.1
Check your Understanding
6.2
Solutions
6.2
Check-Point Questions
6.3
1st-Order Equations
6.3
Check-Point Questions
6.4
2nd-Order Equations
6.4
Check-Point Questions
6.5
Higher-Order Equations
6.5
Check-Point Questions
6.6
Summary & Exercises
6.6
Exercises
7
Undetermined Coefficients
7.1
Nonhomogeneous Equations
7.1
Check-Point Questions
7.2
General Solutions
7.2
Check-Point Questions
7.3
Selecting Particular Solutions
7.3
Check-Point Questions
7.4
Adjusting Particular Solutions
7.4
Check-Point Questions
7.5
Method of Undetermined Coefficients
7.5
Check-Point Questions
7.6
Summary & Exercises
7.6
Exercises
8
Variation of Parameters
8.1
Summary & Exercises
8.1
Exercises
8.2
Orphaned Content
8.2
Additional Practice
IV
Laplace Transform Method
9
Forward Transforms
9.1
Introduction
9.1.1
Motivation
9.1.1
Check-Point Questions
9.1.2
Definition
9.1.2
Check-Point Questions
9.2
Common Transforms
9.2.1
Constant Function,
1
9.2.1
Check-Point Questions
9.2.2
Exponential Function,
e
a
t
9.2.2
Check-Point Questions
9.2.3
Power Function,
t
n
9.2.3
Check-Point Questions
9.2.4
Sine and Cosine,
sin
(
b
t
)
,
cos
(
b
t
)
9.2.4
Check-Point Questions
9.3
Properties of Laplace Transforms
9.3.1
Linearity of the Laplace Transform
9.3.1
Check-Point Questions
9.3.2
Multiplication by
e
a
t
9.3.2
Check-Point Questions
9.3.3
Derivative Transform
9.3.3
Check-Point Questions
9.3.4
Multiplication by
t
n
9.3.4
Check-Point Questions
9.4
Forward Transforming Equations
9.4
Check-Point Questions
9.5
Summary & Exercises
9.5
Exercises
10
Backward Transforms
10.1
Selecting Forms
10.1.1
Matched Forms
10.1.1
Check-Point Questions
10.1.2
Mismatched Forms
10.1.2
Check-Point Questions
10.2
Preparing the Backward Transform
10.2.1
Splitting Fractions
10.2.1
Check-Point Questions
10.2.2
Completing the Square
10.2.2
Check-Point Questions
10.2.3
Partial Fraction Decomposition
10.2.3
Check-Point Questions
10.3
Summary & Exercises
10.3
Exercises
11
Solving Equations
11.1
Overview
11.1
Check-Point Questions
11.2
Step-by-Step Examples
11.2
Check-Point Questions
12
Piecewise Forcing Functions
V
Systems of Equations
13
Linear Systems
14
Nonlinear Systems
15
Applications
VI
Numerical Methods
16
Euler’s Method
17
Runge-Kutta Methods
18
Error Analysis
VII
Orphaned Exercises
19
Miscellaneous Exercises
VIII
Modeling Stuff
20
Intro Modeling
21
SOV Modeling
22
IF Modeling
23
UC Modeling
A
Algebra Review
A.1
Exponential and Logarithmic Functions
A.2
Rational Functions
A.3
Quadratic equations
A.4
Trigonometric Identities
A.5
Solving Polynomial Equations
A.6
Partial Fraction Decomposition
A.6
Exercises
B
Calculus Review
B.1
Product Rule
B.2
Integration by parts
B.2.1
Breaking Down the Integration by Parts Formula
B.2.2
Example: Applying Integration by Parts
B.2.3
Laplace Transform and Integration by Parts: An Analogy
C
Details
C.1
Direct Integration
C.2
Visualizing Solutions
C.3
Integrating Factor
C.4
Linear Homogeneous Constant Coefficients
C.5
Laplace Transforms
D
Orphaned Content (Reader Ignore)
D.1
Orphaned Content
D.1
Additional Practice
D.2
Orphaned Content
D.2
Additional Practice
D.3
Orphaned Content
D.3
Additional Practice
D.4
Orphaned Content
D.4
Additional Practice
Appendix
D
Orphaned Content (Reader Ignore)
D.1
Orphaned Content
D.2
Orphaned Content
D.3
Orphaned Content
D.4
Orphaned Content
