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18.03 Supplementary Notes

Supplementary Notes

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

Full Text of Supplementary Notes (PDF - 1.3 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

Chapter8: 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

Chapter19: 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

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