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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