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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
Section
8.1
Summary & Exercises
Summary of the Key Ideas.
Method
Exercises
Exercises
Do something.
1
.
4
y
″
−
36
y
′
=
0
Answer
.
Answer
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