Differential Equations

Differential Equations

Linear Phase Portraits Mathlet from the d'Arbeloff Interactive Math Project. (Image courtesy of Hu Hohn and Prof. Haynes Miller.)

Course Highlights

This course includes lecture notes, assignments, and a full set of video lectures.

Course Description

Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time. Topics include: Solution of first-order ODE's by analytical, graphical and numerical methods; Linear ODE's, especially second order with constant coefficients; Undetermined coefficients and variation of parameters; Sinusoidal and exponential signals: oscillations, damping, resonance; Complex numbers and exponentials; Fourier series, periodic solutions; Delta functions, convolution, and Laplace transform methods; Matrix and first order linear systems: eigenvalues and eigenvectors; and Non-linear autonomous systems: critical point analysis and phase plane diagrams.

Special Features

• Video lectures

Lecture Notes

Below are the lecture notes for every lecture session. Some lecture sessions also have supplementary files called "Muddy Card Responses." Students pick up half pages of scrap paper when they come into the classroom, jot down on them what they found to be the most confusing point in the day's lecture or the question they would have liked to ask. Responses are then made available to all students on the class Web site.

SES #	TOPICS
I. First-order Differential Equations
L0	Simple Models and Separable Equations
L1	Direction Fields, Existence and Uniqueness of Solutions (PDF)
L2	Numerical Methods (PDF)
L3	Linear Equations: Models (PDF)
L4	Solution of Linear Equations, Variation of Parameter (PDF)
L5	Complex Numbers, Complex Exponentials (PDF)
L6	Roots of Unity; Sinusoidal Functions (PDF)
L7	Linear System Response to Exponential and Sinusoidal Input; Gain, Phase Lag (PDF)
L8	Autonomous Equations; The Phase Line, Stability (PDF) Muddy Card Responses (PDF)
L9	Linear vs. Nonlinear (PDF)
L10	Hour Exam I
II. Second-order Linear Equations
L11	The Spring-mass-dashpot Model; Superposition Characteristic Polynomial; Real Roots; Initial Conditions (PDF) Muddy Card Responses (PDF)
L12	Complex Roots; Damping Conditions (PDF)
L13	Inhomogeneous Equations, Superposition (PDF)
L14	Operators and Exponential Signals (PDF) Muddy Card Responses (PDF)
L15	Undetermined Coefficients (PDF)
L16	Frequency Response (PDF)
L17	Applications: Guest appearance by EECS Professor Jeff Lang (PDF) Supplementary Notes Driving Through the Dashpot (PDF)
L18	Exponential Shift Law; Resonance (PDF)
L19	Hour Exam II
III. Fourier Series
L20	Fourier Series (PDF)
L21	Operations on Fourier series (PDF) Muddy Card Responses (PDF)
L22	Periodic Solutions; Resonance (PDF)
IV. The Laplace Transform
L23	Step Function and delta Function (PDF)
L24	Step Response, Impulse Response (PDF)
L25	Convolution (PDF)
L26	Laplace Transform: Basic Properties (PDF) Muddy Card Responses (PDF)
L27	Application to ODEs; Partial Fractions (PDF)
L28	Completing the Square; Time Translated Functions (PDF) Muddy Card Responses (PDF)
L29	Pole Diagram (PDF)
L30	Hour Exam III
V. First Order Systems
L31	Linear Systems and Matrices (PDF)
L32	Eigenvalues, Eigenvectors (PDF)
L33	Complex or Repeated Eigenvalues (PDF)
L34	Qualitative Behavior of Linear Systems; Phase Plane (PDF)
L35	Normal Modes and the Matrix Exponential (PDF)
L36	Inhomogeneous Equations: Variation of Parameters Again (PDF) Muddy Card Responses (PDF)
L37	Nonlinear Systems (PDF)
L38	Examples of Nonlinear Systems (PDF)
L39	Chaos (PDF)
L40	Final Exam

Video Lectures

These video lectures of Professor Arthur Mattuck teaching 18.03 were recorded live in the Spring 2003 and do not correspond precisely to the lectures taught in the Spring of 2006. Professor Mattuck has inspired and informed generations of MIT students with his engaging lectures.

The videotaping was made possible by The d'Arbeloff Fund for Excellence in MIT Education.

Note: Lecture 18, 34, and 35 are not available.

LEC #	TOPICS
1	The Geometrical View of y'=f(x,y): Direction Fields, Integral Curves.
2	Euler's Numerical Method for y'=f(x,y) and its Generalizations.
3	Solving First-order Linear ODE's; Steady-state and Transient Solutions.
4	First-order Substitution Methods: Bernouilli and Homogeneous ODE's.
5	First-order Autonomous ODE's: Qualitative Methods, Applications.
6	Complex Numbers and Complex Exponentials.
7	First-order Linear with Constant Coefficients: Behavior of Solutions, Use of Complex Methods.
8	Continuation; Applications to Temperature, Mixing, RC-circuit, Decay, and Growth Models.
9	Solving Second-order Linear ODE's with Constant Coefficients: The Three Cases.
10	Continuation: Complex Characteristic Roots; Undamped and Damped Oscillations.
11	Theory of General Second-order Linear Homogeneous ODE's: Superposition, Uniqueness, Wronskians.
12	Continuation: General Theory for Inhomogeneous ODE's. Stability Criteria for the Constant-coefficient ODE's.
13	Finding Particular Sto Inhomogeneous ODE's: Operator and Solution Formulas Involving Ixponentials.
14	Interpretation of the Exceptional Case: Resonance.
15	Introduction to Fourier Series; Basic Formulas for Period 2(pi).
16	Continuation: More General Periods; Even and Odd Functions; Periodic Extension.
17	Finding Particular Solutions via Fourier Series; Resonant Terms; Hearing Musical Sounds.
19	Introduction to the Laplace Transform; Basic Formulas.
20	Derivative Formulas; Using the Laplace Transform to Solve Linear ODE's.
21	Convolution Formula: Proof, Connection with Laplace Transform, Application to Physical Problems.
22	Using Laplace Transform to Solve ODE's with Discontinuous Inputs.
23	Use with Impulse Inputs; Dirac Delta Function, Weight and Transfer Functions.
24	Introduction to First-order Systems of ODE's; Solution by Elimination, Geometric Interpretation of a System.
25	Homogeneous Linear Systems with Constant Coefficients: Solution via Matrix Eigenvalues (Real and Distinct Case).
26	Continuation: Repeated Real Eigenvalues, Complex Eigenvalues.
27	Sketching Solutions of 2x2 Homogeneous Linear System with Constant Coefficients.
28	Matrix Methods for Inhomogeneous Systems: Theory, Fundamental Matrix, Variation of Parameters.
29	Matrix Exponentials; Application to Solving Systems.
30	Decoupling Linear Systems with Constant Coefficients.
31	Non-linear Autonomous Systems: Finding the Critical Points and Sketching Trajectories; the Non-linear Pendulum.
32	Limit Cycles: Existence and Non-existence Criteria.
33	Relation Between Non-linear Systems and First-order ODE's; Structural Stability of a System, Borderline Sketching Cases; Illustrations Using Volterra's Equation and Principle.