18.03 | Spring 2010 | Undergraduate

# Differential Equations

## 18.03 Supplementary Notes

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)

### 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
8.3. 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 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 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 16: More on Fourier Series (PDF)

16.1. Symmetry and Fourier Series
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 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)