18.03 | Spring 2010 | Undergraduate

# Differential Equations

Listed in the table below are reading assignments for each lecture session.

[EP] refers to the course textbook: Edwards, C., and D. Penney. Elementary Differential Equations with Boundary Value Problems. 6th ed. Upper Saddle River, NJ: Prentice Hall, 2003. ISBN: 9780136006138

[SN] refers to the “18.03 Supplementary Notes” written by Prof. Miller.

[Notes] refers to the “18.03 Notes and Exercises” written by Prof. Mattuck.

Also listed are links to specially written Java™ applets, or Mathlets, that were used in the lecture session.

SES # TOPICS READINGS AND RELATED MATHLETS
I. First-order differential equations
R1 Natural growth, separable equations

[EP]: 1.1 and 1.4

L1 Direction fields, existence and uniqueness of solutions

[EP]: 1.2 and 1.3

[Notes]: G.1 (PDF)

[SN]: 1 (PDF)

Isoclines Mathlet

L2 Numerical methods [EP]: 6.1 and 6.2

[Notes]: G.2 (PDF)

Euler’s method Mathlet

L3 Linear equations, models [EP]: 1.5

[SN]: 2 (PDF)

L4 Solution of linear equations, integrating factors [EP]: 1.5

[SN]: 3 (PDF)

L5 Complex numbers, roots of unity [SN]: 5 (PDF)

[SN]: 6 (PDF)

[Notes]: C.1-3 (PDF)

L6 Complex exponentials; sinusoidal functions [SN]: 4 (PDF)

[Notes]: C.4 (PDF) and IR.6 (PDF)

Complex roots Mathlet

Complex exponential Mathlet

L7 Linear system response to exponential and sinusoidal input; gain, phase lag [SN]: 4 (PDF)

[Notes]: IR.6 (PDF)

Trigonometric identity Mathlet

L8 Autonomous equations; the phase line, stability [EP]: 1.7 and 7.1

[SN]: Appendix A (PDF)

Phase lines Mathlet

L9 Linear vs. nonlinear [SN]: Appendix C (PDF)
II. Second-order linear equations
L11 Modes and the characteristic polynomial [EP]: 2.1, 2.2, and 2.3 up to “Polynomial Operators”

[SN]: 9 (PDF)

L12 Good vibrations, damping conditions [EP]: 2.3 and 2.4

Damped vibrations Mathlet

L13 Exponential response formula, spring drive [EP]: 2.6, pp. 157-159

[SN]: 7 (PDF) (for beats)

[Notes]: O.1 (PDF)

L14 Complex gain, dashpot drive [EP]: 2.6, pp. 165-167

[SN]: 10 (PDF)

[Notes]: O.1, 2, 4 (PDF)

L15 Operators, undetermined coefficients, resonance

[EP]: 2.5, pp. 144-153 and EP: 2.7

[SN]: 8 (PDF)

[SN]: 11 (PDF)

L16 Frequency response [SN]: 14 (PDF)

L17 LTI systems, superposition, RLC circuits. [SN]: Appendix B (PDF)

Series RLC circuit Mathlet

L18 Engineering applications [SN]: 12 (PDF)

[SN]: 13 (PDF)

[Notes]: O.3 (PDF)

III. Fourier series
L20 Fourier series [EP]: 8.1

[SN]: 16 (PDF)

Fourier coefficients Mathlet

L21 Operations on fourier series [EP]: 8.2 and 8.3

L22 Periodic solutions; resonance [EP]: 8.3 and 8.4
L23 Step function and delta function [SN]: 17 (PDF)
L24 Step response, impulse response [SN]: 18 (PDF)

[Notes]: IR (PDF)

L25 Convolution [SN]: 18 (PDF)

Convolution: Accumulation Mathlet

Convolution: Flip and drag Mathlet

L26 Laplace transform: basic properties [EP]: 4.1
L27 Application to ODEs [EP]: 4.2 and 4.3

[SN]: 20 (PDF)

[Notes]: H

L28 Second order equations; completing the squares [EP]: 4.5 and 4.6

[SN]: 20 (PDF)

L29 The pole diagram [EP]: 4.4

[SN]: 22 (PDF)

[SN]: 23 (PDF)

18.03 Difference Equations and Z-Transforms (PDF)(Courtesy of Jeremy Orloff.)

Poles and vibrations Mathlet

L30 The Transfer function and frequency response
IV. First order systems
L32 Linear systems and matrices [EP]: 5.1-5.3

[SN]: 25 (PDF)

[Notes]: LS.1 (PDF)

L33 Eigenvalues, eigenvectors [EP]: 5.4

[Notes]: LS.2 (PDF)

Matrix vector Mathlet

L34 Complex or repeated eigenvalues [EP]: 5.4

[Notes]: LS.3 (PDF)

L35 Qualitative behavior of linear systems; phase plane [SN]: 26 (PDF)

L36 Normal modes and the matrix exponential [EP]: 5.7

[Notes]: LS.6 (PDF)

L37 Nonlinear systems [EP]: 7.2 and 7.3

[Notes]: LS.6 (PDF)

L38 Linearization near equilibria; the nonlinear pendulum [EP]: 7.4 and 7.5

[Notes]: GS (PDF)

[SN]: Appendix B (PDF)

[SN]: Appendix C (PDF)

L39 Limitations of the linear: limit cycles and chaos Vector fields Mathlet

These notes and exercises were written by Prof. Arthur Mattuck and are designed to supplement the textbook.

SECTIONS TOPICS
Notes
D Definite integral solutions (PDF)
G Graphical and numerical methods (PDF)
C Complex numbers (PDF)
IR Input-response models (PDF)
O Linear differential operators (PDF)
S Stability (PDF)
I Impulse response and convolution (PDF)
LT Laplace transform (PDF)
LS Linear systems of ODE’s - LS1 of 6 (PDF), LS2 of 6 (PDF), LS3 of 6 (PDF), LS4 of 6 (PDF), LS5 of 6 (PDF), LS6 of 6 (PDF)
GS Graphing systems (PDF)
LC Limit cycles (PDF)
Exercises
1 First-order ODE’s (PDF)
2 Higher-order ODE’s (PDF)
3 Laplace transform (PDF)
4 Linear systems (PDF)
5 Graphing systems (PDF)
6 Power series (PDF)
7 Fourier series (PDF)
Solutions to exercises
1 First-order ODE’s (PDF)
2 Higher-order ODE’s (PDF)
3 Laplace transform (PDF)
4 Linear systems (PDF)
5 Graphing systems (PDF)
6 Power series (PDF)
7 Fourier series (PDF)

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
8.2. 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 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
16.2. 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 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)

## Course Info

Spring 2010
##### Learning Resource Types
Lecture Videos
Simulations
Lecture Notes
Problem Sets with Solutions