MIT Concourse 18.03 - Ordinary Differential Equations - Supplementary Materials

18.03 Notes (Arthur Mattuck) Exercises and Solutions

D. Definite Integral Solutions

G. Graphical and Numerical Methods

C. Complex Numbers

IR. Input Response Models

O. Linear Differential Operators

S. Stability

I. Impulse Response and Convolution

H. Heaviside Coverup Method

LT. Laplace Transform

CG. Convolution and Green's Formula   

LS.1. Linear Systems: Review of Linear Algebra

LS.2. Homogeneous Linear Systems with Constant Coefficients

LS.3. Complex and Repeated Eigenvalues

LS.4. Decoupling Systems

LS.5. Theory of Linear Systems

LS.6. Solution Matrices

GS.1-6. Graphing ODE Systems

GS.7-8. Structural stability

LC. Limit Cycles

FR. Frequency Response

P. Poles and Amplitude Response

1. First Order ODE's   Solutions

2. Higher Order ODE's   Solutions

3. Laplace Transform   Solutions

4. Linear Systems   Solutions

5. Graphing Systems   Solutions

6. Power Series   Solutions

7. Fourier Series   Solutions

8. Extra Problems   Solutions

Supplementary Notes (Haynes Miller - Spring 2012 versions)

#0. Preface

#1. Notation and language

#2. Modeling by first order linear ODEs

#3. Solutions of first order linear ODEs

#4. Sinusoidal solutions

#5. The algebra of complex numbers

#6. The complex exponential

#7. Beats

#8. RLC circuits

#9. Normalization of solutions

#10. Operators and the exponential response formula

#11. Undetermined coefficients

#12. Resonance

#13. Time invariance

#14. The exponential shift law

#15. Natural frequency and damping ratio

#16. Frequency response

#17 Resonance, not: the Tacomah Narrows Bridge

#18 Linearization: the phugoid equation as example

#19 The Wronskian

#20 More on Fourier series

#21 Steps, impulses and generalized functions

#22 Generalized functions and differential equations

#23 Impulse and step responses

#24 Convolution

#25 Laplace Transform technique: coverup

#26 The Laplace Transform and generalized functions

#27 The pole diagram and the Laplace Transform

#28 Amplitude response and the pole diagram

#29 The Laplace Transform and more general systems

#30 First order systems and second order equations

#31 Phase portraits in two dimensions

Concourse 18.03 Linear Algebra Supplements (Robert Winters, Spring 2014)

Systems of Linear Equations and Row Reduction

Matrices and Vectors – Meaning of Columns of a Matrix

Matrix Algebra and Inverse Matrices

Subspaces, Span, Linear Independence, Basis of a Subspace; Images and Kernel of a Matrix

Coordinates Relative to a Basis; Matrix of a Linear Transformation Relative to Bases

General Linear Spaces (Vector Spaces) and Solutions of ODEs

A Worked Example of a 2nd Order Linear System with Sinusoidal Input Signal

Laplace Transform Facts

Worked Examples of Laplace Transform and Convolution

Notes on Convolution

Sampler of Phase Plane diagrams

Supplement on Evolution Matrices

Supplement on Linear Coordinates, Alternate Bases, and Evolution Matrices

Matrix Methods for Solving Systems of 1st Order Linear Differential Equations

Nonlinear Systems and Linearization

Concourse 18.03 Lecture Notes (Robert Winters)

Lecture #1 Notes (revised Feb 2, 2020)

Lecture #2 Notes (may be revised)

Lecture #3 Notes (may be revised)

Lecture #4 Notes (may be revised)

Lecture #5 Notes (may be revised)

Lecture #6-7 Notes (may be revised)

Lecture #8 Notes (may be revised)

Lecture #9 Notes (may be revised)

Lecture #10 Notes (revised Mar 17, 2018)

Lecture #11 Notes (may be revised)

Lecture #12 Notes (may be revised)

Lecture #13 Notes (may be revised)

Lecture #14 Notes (may be revised)

Lecture #15 Notes (new, Apr 7, 2019)

Lecture #16 Notes (new, Apr 21, 2019)

Lecture #17-18 Notes (revised, May 27, 2019)

18.03 Supplementary Notes (Haynes Miller, Spring 2010)

These notes (Spring 2010 version) were written by Prof. Haynes Miller and were designed to supplement the Edwards & Penney 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)

Notes ©Arthur Mattuck and M.I.T. 1988, 1992, 1996, 2003, 2007.
Suppementary Notes ©Haynes Miller and M.I.T. 2004, 2006, 2008, and 2010.
Concourse 18.03 Lecture Notes ©Robert Winters 2013-2019.