Supplementary Notes

These notes are written by Prof. Haynes Miller and are designed to supplement the textbook.

Full Text of Supplementary Notes (PDF - 1.2 MB)

Preface (PDF)

Chapter 1: Notation and Language (PDF)

1.1. Numbers
1.2. Dependent and Independent Variables
1.3. Equations and Parametrizations
1.4. Parametrizing the Set of Solutions of a Differential Equation
1.5. Solutions of ODEs

Chapter 2: Modeling by First Order Linear ODEs (PDF)

2.1. The Savings Account Model
2.2. Linear Insulation
2.3. System, Signal, System Response

Chapter 3: Solutions of First Order Linear ODEs (PDF)

3.1. Homogeneous and Inhomogeneous; Superposition
3.2. Variation of Parameters
3.3. Continuation of Solutions
3.4. Final Comments on the Bank Account Model

Chapter 4: Sinusoidal Solutions (PDF)

4.1. Periodic and Sinusoidal Functions
4.2. Periodic Solutions and Transients
4.3. Amplitude and Phase Response

Chapter 5: The Algebra of Complex Numbers (PDF)

5.1. Complex Algebra
5.2. Conjugation and Modulus
5.3. The Fundamental Theorem of Algebra

Chapter 6: The Complex Exponential (PDF)

6.1. Exponential Solutions
6.2. The Complex Exponential
6.3. Polar Coordinates
6.4. Multiplication
6.5. Roots of Unity and Other Numbers

Chapter 7: Beats (PDF)

7.1. What Beats Are
7.2. What Beats Are Not

Chapter 8: Linearization: The Phugoid Equation as Example (PDF)

8.1. The Airplane System Near Equilibrium
8.2. Deriving the Linearized Equation of Motion
8.3. Implications

Chapter 9: Normalization of Solutions (PDF)

9.1. Initial Conditions
9.2. Normalized Solutions
9.3. More on Hyperbolic Functions
9.4. ZSR/ZIR

Chapter 10: Operators and the Exponential Response Formula (PDF)

10.1. Operators
10.2. LTI Operators and Exponential Signals
10.3. Real and Complex Solutions

Chapter 11: Undetermined Coefficients (PDF)

Chapter 12: Resonance and the Exponential Shift Law (PDF)

12.1. Exponential Shift
12.2. Product Signals
12.3. Resonance
12.4. Higher Order Resonance
12.5. Summary

Chapter 13: Natural Frequency and Damping Ratio (PDF)

Chapter 14: Filters and Frequency Response (PDF)

Chapter 15: The Wronskian (PDF)

Chapter 16: Impulses and Generalized Functions (PDF)

16.1. From Bank Accounts to the Delta Function
16.2. The Delta Function
16.3. Integrating Generalized Functions
16.4. The Generalized Derivative

Chapter 17: Impulse and Step Responses (PDF)

17.1. Impulse Response
17.2. Impulses in Second Order Equations
17.3. Singularity Matching
17.4. Step Response

Chapter 18: Convolution (PDF)

18.1. Superposition of Infinitesimals: The Convolution Integral
18.2. Example: The Build Up of a Pollutant in a Lake
18.3. Convolution as a Product

Chapter 19: Laplace Transform Technique: Cover-up (PDF)

19.1. The "Cover-up Method"
19.2. Laplace Transform of Impulse and Step Responses
19.3. List of Properties of the Laplace Transform

Chapter 20: The Pole Diagram and the Laplace Transform (PDF)

20.1. Poles and the Pole Diagram
20.2. The Pole Diagram of the Laplace Transform
20.3. The Laplace Transform Integral
20.4. Transforms of Periodic Functions

Chapter 21: The Laplace Transform and Generalized Functions (PDF)

21.1. What the Laplace Transform Doesn’t Tell Us
21.2. Worrying about t = 0
21.3. The t-derivative Rule
21.4. The Initial Singularity Formula
21.5. The Initial Value Formula
21.6. Final Value Formula
21.7. Laplace Transform of the Unit Impulse Response
21.8. Initial Conditions

Chapter 22: The Laplace Transform and more General Systems (PDF)

22.1. Zeros of the Laplace Transform: Stillness in Motion
22.2. General LTI Systems

Chapter 23: More on Fourier Series (PDF)

23.1. Harmonic Response
23.2. The Gibbs Phenomenon
23.3. Symmetry and Fourier Series
23.4. Symmetry about Other Points
23.5. Fourier Distance
23.6. Complex Fourier Series
23.7. Laplace Transform and Fourier Series

Chapter 24: First Order Systems and Second Order Equations (PDF)

24.1. The Companion System
24.2. Initial Value Problems

Chapter 25: Phase Portraits in Two Dimensions (PDF)

25.1. Phase Portraits and Eigenvectors
25.2. The (tr, det) Plane and Structural Stability
25.3. The Portrait Gallery