Ordinary Differential Equations: The Integrated Problem-Solving Approach

Preface 
Acknowledgments
 

Part 1 Preliminary Topics

Chapter 1 Before Differentiation 
1.1 Zeros of Polynomials 
1.2 Implicitly Defined Functions 
1.3 The Functions e^x and ln x
1.4 Cramer’s Formulas 
1.5 Linear Algebra Notations 

Chapter 2 Differentiation 
2.1 Functions of a Single Variable 
2.2 Functions of Two Variables

Chapter 3 Integration 
3.1 Basic Rules of Integration 
        3.1.1 The Substitution Rule 
        3.1.2 Integration by Parts 
3.2 Integration of Rational Functions 
        3.2.1 The Three Algebric Steps 
        3.2.2 The Six Types of Integrals 
        3.2.3 More Examples of Integration of Rational Functions 
3.3 Transformations to Integration of Rational Functions 
        3.3.1 Integrals Like ∫ R (e^x) dx, where R Is a Rational Function 
        3.3.2 Euler Substitutions 
        3.3.3 More Examples of Integration by Parts Rule 

Part 2 The Lectures

Chapter 4 Lecture 1. Separable and Linear (n = 1) Equations 
4.1 What Is a Differential Equation? The First Examples 
4.2 Separable Differential Equation
4.3 Linear (Order One) Differential Equation 
Summary 

Chapter 5 Lecture 2. Bernoulli and Riccati Equations 
5.1 Bernoulli Differential Equations 
5.2 Riccati Differential Equations 

Chapter 6 Lecture 3. Equations with y′′ That Can Be Reduced to Two Order One Equations 
6.1 Basic Terminology 
6.2 No y Equations 
6.3 No x Equations 
6.4 Equations with Both x and y Absent 

Chapter 7 Lecture 4. Exact Differential Equation and Integrating Factors 
7.1 Exact Differential Equation 
7.2 Integrating Factors of a Single Variable 
7.3 The Formula for Linear (Order One) Differential Equations 

Chapter 8 Lecture 5. Some Applications 
8.1 Basic Terminology 
8.2 Orthogonal Trajectories
8.3 Clairaut’s Equation 
8.4 The Brachistochrone Problem 

Chapter 9 Lecture 6. Homogeneous, Linear Differential Equations 
9.1 Basic Terminology 
9.2 n = 2 General Solution for Homogeneous Equations 
9.3 n = 2 Initial Value Problem for Homogeneous Equations 
9.4 n = 3 Initial Value Problem for Homogeneous Equations 

Chapter 10 Lecture 7. Undetermined Coefficients Method 
10.1 Basic Terminology 
10.2 Undetermined Coefficients Method. General Solution 
10.3 Undetermined Coefficients Method. Initial Value Problem 
Looking Ahead 

Chapter 11 Lecture 8. Variation of Parameters Method 
11.1 Basic Terminology 
11.2 Examples 

Chapter 12 Lecture 9. Cauchy–Euler Equations 
12.1 Basic Terminology 
12.2 Examples 

Chapter 13 Lecture 10. Using Power Series With Differential Equations 
13.1 Basic Terminology 
13.2 Two Ways to Define an Infinite Sequence of Numbers 
13.3 Simple Examples 
13.4 A More Complicated Example 

Chapter 14 Lecture 11. Systems of Linear Differential Equations 
14.1 Basic Terminology 
14.2 From Equation of Order n to the Corresponding n n × System 
14.3 The Case of Two Different Eigenvalues 
14.4 The Case of Multiplicity Two Eigenvalues 
14.5 The Case of a Pair of Complex Conjugate Eigenvalues 

Chapter 15 Lecture 12. IVP for n = 2 Systems of Equations 
15.1 Basic Terminology 
15.2 Case 1: Repeated Eigenvalue (Multiplicity 2) 
15.3 Case 2: Complex Conjugate Eigenvalues 

Chapter 16 Lecture 13. n = 3 Systems of Equations 
16.1 Basic Terminology 
16.2 Finding the Eigenvalues for a 3 3 × Matrix 
16.3 Using the “rref” Command on a TI-84 Calculator 

Chapter 17 Lecture 14. Systems of Three Equations. Simple Eigenvalues 
17.1 Three Distinct Real Eigenvalues 
17.2 Complex Conjugate Eigenvalues 

Chapter 18 Lecture 15. Systems of Three Equations. Repeated Eigenvalues 
18.1 Basic Terminology 
18.2 Case 1: Eigenvalue of Multiplicity 2 
18.3 Case 2: Eigenvalue of Multiplicity 3