These notes were written by Prof. Haynes Miller and are designed to supplement the textbook. They are available as individual chapters below or compiled into a complete set. (PDF - 1.5MB)

### Preface (PDF)

### Chapter 1: Notation and Language (PDF)

1.1. Dependent and Independent Variables

1.2. 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: RLC Circuits (PDF)

8.1. Series RLC Circuits

8.2. A Word About Units

8.3. Implications

### Chapter 9: Normalization of Solutions (PDF)

9.1. Initial Conditions

9.2. Normalized Solutions

9.3. ZSR/ZIR

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

10.1. Operators

10.2. LTI Operators and Exponential Signals

10.3. Sinusoidal Signals

10.4. Damped Sinusoidal Signals

10.5. Time Invariance

### 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: Frequency Response (PDF)

14.1. Driving Through the Spring

14.2. Driving Through the Dashpot

14.3. Second Order Frequency Response Using Damping Ratio

### Chapter 15: The Wronskian (PDF)

### Chapter 16: More on Fourier Series (PDF)

16.1. Symmetry and Fourier Series

16.2. Symmetry about Other Points

16.3 The Gibbs Effect

16.4. Fourier Distance

16.5. Complex Fourier Series

16.6 Harmonic Response

### Chapter 17: Impulses and Generalized Functions (PDF)

17.1. From Bank Accounts to the Delta Function

17.2. The Delta Function

17.3. Integrating Generalized Functions

17.4. The Generalized Derivative

### Chapter 18: Impulse and Step Responses (PDF)

18.1. Impulse Response

18.2. Impulses in Second Order Equations

18.3. Singularity Matching

81.4. Step Response

### Chapter 19: Convolution (PDF)

19.1. Superposition of Infinitesimals: The Convolution Integral

19.2. Example: The Build Up of a Pollutant in a Lake

19.3. Convolution as a Product

### Chapter 20: Laplace Transform Technique: Cover-up (PDF)

20.1. Simple Case

20.2. Repeated Roots

20.3. Completing The Square. Suppose

20.4. Complex Coverup

20.5. Complete PArtial Fractions

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

21.1. Laplace Transform of Impulse and Step Responses

21.2. What the Laplace Transform Doesn’t Tell Us

21.3. Worrying about t = 0

21.4. The t-derivative Rule

21.5. The Initial Singularity Formula

21.7. The Initial Value Formula

21.8. Initial Conditions

### Chapter 22: The Pole Diagram and the Laplace Transform (PDF)

22.1. Poles and the Pole Diagram

22.2. The Pole Diagram of the Laplace Transform

22.3. The Laplace Transform Integral

22.4. TranLaplace Transform

### Chapter 23: Amplitude Response and the Pole Diagram (PDF)

### Chapter 24: The Laplace Transform and more General Systems (PDF)

22.1. Zeros of the Laplace Transform: Stillness in Motion

22.2. General LTI Systems

### Chapter 25: First Order Systems and Second Order Equations (PDF)

25.1. The Companion System

25.2. Initial Value Problems

### Chapter 26: Phase Portraits in Two Dimensions (PDF)

26.1. Phase Portraits and Eigenvectors

26.2. The (tr, det) Plane and Structural Stability

26.3. The Portrait Gallery

### Appendices

Appendix A. The Kermack-McKendrick Equation (PDF)

Appendix B. The Tacoma Narrows Bridge: Resonance vs Flutter (PDF)

Appendix C. Linearization: The Phugoid Equation as Example (PDF)