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Differential Equations Notes PDF

Differential Equations

Screenshot of Mathlet from the d'Arbeloff Interactive Math Project.
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

Technical Requirements

Special software is required to use some of the files in this course: .jar.

Important Notes : -

It is a collection of lectures notes not ours. Our subjective is to help students to find all engineering notes with different lectures PowerPoint slides in ppt ,pdf or html file at one place. Because we always face that we lose much time by searching in Google or yahoo like search engines to find or downloading a good lecture notes in our subject area with free. Also it is difficult to find popular authoress or books slides with free of cost. If you find any copyrighted slides or notes then please inform me immediately by comments or email as following address .I will take actions to remove it. Please click bellow to download ppt slides/ pdf notes. If you face any problem in downloading or if you find any link not correctly work or if you have any idea to improve this blog/site or if you find any written mistake or you think some subjects notes should be include then give your suggestion as comment by clicking on comment link bellow the post (bottom of page) or email us in this address on I will must consider your comments only within 1-2 days. if you have any good class notes/lecture slides in ppt or pdf or html format then please you upload these files to rapidshare.come and send us links or all files by our email address on

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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.

I. First-order Differential Equations
L0Simple Models and Separable Equations
L1Direction Fields, Existence and Uniqueness of Solutions (PDF)
L2Numerical Methods (PDF)
L3Linear Equations: Models (PDF)
L4Solution of Linear Equations, Variation of Parameter (PDF)
L5Complex Numbers, Complex Exponentials (PDF)
L6Roots of Unity; Sinusoidal Functions (PDF)
L7Linear System Response to Exponential and Sinusoidal Input; Gain, Phase Lag (PDF)
L8Autonomous Equations; The Phase Line, Stability (PDF) Muddy Card Responses (PDF)
L9Linear vs. Nonlinear (PDF)
L10Hour Exam I
II. Second-order Linear Equations
L11The Spring-mass-dashpot Model; Superposition Characteristic Polynomial; Real Roots; Initial Conditions (PDF) Muddy Card Responses (PDF)
L12Complex Roots; Damping Conditions (PDF)
L13Inhomogeneous Equations, Superposition (PDF)
L14Operators and Exponential Signals (PDF) Muddy Card Responses (PDF)
L15Undetermined Coefficients (PDF)
L16Frequency Response (PDF)
L17Applications: Guest appearance by EECS Professor Jeff Lang (PDF) Supplementary Notes Driving Through the Dashpot (PDF)
L18Exponential Shift Law; Resonance (PDF)
L19Hour Exam II
III. Fourier Series
L20Fourier Series (PDF)
L21Operations on Fourier series (PDF) Muddy Card Responses (PDF)
L22Periodic Solutions; Resonance (PDF)
IV. The Laplace Transform
L23Step Function and delta Function (PDF)
L24Step Response, Impulse Response (PDF)
L25Convolution (PDF)
L26Laplace Transform: Basic Properties (PDF) Muddy Card Responses (PDF)
L27Application to ODEs; Partial Fractions (PDF)
L28Completing the Square; Time Translated Functions (PDF) Muddy Card Responses (PDF)
L29Pole Diagram (PDF)
L30Hour Exam III
V. First Order Systems
L31Linear Systems and Matrices (PDF)
L32Eigenvalues, Eigenvectors (PDF)
L33Complex or Repeated Eigenvalues (PDF)
L34Qualitative Behavior of Linear Systems; Phase Plane (PDF)
L35Normal Modes and the Matrix Exponential (PDF)
L36Inhomogeneous Equations: Variation of Parameters Again (PDF) Muddy Card Responses (PDF)
L37Nonlinear Systems (PDF)
L38Examples of Nonlinear Systems (PDF)
L39Chaos (PDF)
L40Final 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.

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

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