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)

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


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)

Course Info

As Taught In
Spring 2010
Learning Resource Types
Lecture Videos
Lecture Notes
Problem Sets with Solutions