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.

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)

Harmonic frequency response: Variable input frequency Mathlet

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

[SN]: 10 (PDF)

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

Amplitude and phase: Second order II Mathlet

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)

Amplitude and phase: Second order II Mathlet

Amplitude and phase: First order Mathlet

Amplitude and phase: Second order III Mathlet

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

Fourier coefficients: Complex with sound Mathlet

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.)

Amplitude response: Pole diagram Mathlet

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)

Linear phase portrait: Matrix entry Mathlet

Matrix vector Mathlet

L34 Complex or repeated eigenvalues [EP]: 5.4

[Notes]: LS.3 (PDF)

Linear phase portrait: Matrix entry Mathlet

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

Linear phase portrait: Matrix entry Mathlet

Linear phase portrait: Cursor entry Mathlet

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.

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)
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)

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