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Course18.03SC122 lectures

Differential Equations

Prof. Arthur Mattuck and others · Massachusetts Institute of Technology

Start with lecture 1 At MIT OpenCourseWare

About

The laws of nature are expressed as differential equations. Scientists and engineers must know how to model the world in terms of differential equations, and how to solve those equations and interpret the solutions. This course focuses on the equations and techniques most useful in science and engineering.

Course Format

This course has been designed for independent study. It provides everything you will need to understand the concepts covered in the course. The materials include:

Lecture Videos by Professor Arthur Mattuck.

Course Notes on every topic.

Practice Problems with Solutions.

Problem Solving Videos taught by experienced MIT Recitation Instructors.

Problem Sets to do on your own with Solutions to check your answers against when you’re done.

A selection of Interactive Java® Demonstrations called Mathlets to illustrate key concepts.

A full set of Exams with Solutions, including practice exams to help you prepare.

Content Development

Haynes Miller

Lectures

Basic DE's and Separable Equations

  1. Introduction to the 18.03 Applets 9 min · Video
  2. Separable Equations 8 min · Video

Geometric Methods

  1. Geometric View of DE's: Direction Fields, Integral Curves Video
  2. Example, Existence and Uniqueness Video
  3. Example Where Existence and Uniqueness Fails Video
  4. Exploration of the Isoclines Applet 11 min · Video
  5. Direction Fields 12 min · Video

Numerical Methods

  1. Euler's Method Video
  2. Example of Euler's Method Video
  3. Exploration of the Euler's Method Applet 7 min · Video
  4. Better Methods Video
  5. Pitfalls Video
  6. Euler's Method 11 min · Video

First Order Linear ODE's

  1. Linear First Order ODE's: Definition Video
  2. Linear First Order ODE's: Examples Video
  3. Linear Equations 11 min · Video

Solution by Integrating Factors

  1. Solutions to Linear First Order ODE's Video
  2. Solutions of First Order Linear Equations 9 min · Video

Complex Arithmetic and Exponentials

  1. Complex Arithmetic, Euler's Formula Video
  2. Justification of Euler's Formula Video
  3. Polar Form Video
  4. Example: Complex Replacement Video
  5. Nth Roots Video
  6. Complex Numbers and Euler's Formula 12 min · Video

Sinusoidal Functions

  1. Three Proofs of the Basic Trigonometric Identity Video
  2. Sinusoidal Functions 16 min · Video

First Order Constant Coefficient Linear ODE's

  1. Solutions to Constant Coefficient Linear First Order ODE's Video
  2. Examples of Constant Coefficient Linear First Order ODE's Video
  3. More Examples of Constant Coefficient Equations Video
  4. First Order Constant Coefficient Linear ODE's 14 min · Video

Exponential Input; Gain and Phase Lag

  1. Exploration of the Amplitude and Phase: First Order Applet 9 min · Video
  2. Sinusoidal Inputs 14 min · Video

First Order Autonomous Differential Equations

  1. First Order Autonomous Equations Video
  2. Bank Account Video
  3. Example: Logistic Model Video
  4. Exploration of the Phase Lines Applet 9 min · Video
  5. Example: Logistic Model with Harvesting Video
  6. Autonomous Equations and Phase Lines 12 min · Video

Modes and the Characteristic Equation

  1. Constant Coefficient Second Order Homogeneous DE's Video
  2. Real Roots: Overdamping Video
  3. Complex Roots: Underdamping Video
  4. Repeated Roots: Critical Damping Video
  5. Real Solutions from Complex Roots Video
  6. Homogeneous Constant Coefficient Equations: Real Roots 10 min · Video
  7. Homogeneous Constant Coefficient Equations: Any Roots 12 min · Video

Damped Harmonic Oscillators

  1. Damping and Pseudo-Frequency Video
  2. Exploration of the Damped Vibrations Applet 7 min · Video
  3. Damped Harmonic Oscillators 10 min · Video

Exponential Response

  1. Linear Operators and Superposition 51 min · Video
  2. Inhomogeneous Second Order DE's: Definitions and Models Video
  3. Particular and General Solutions Video
  4. Inhomogeneous DE's with Exponential Input Video
  5. Forced Oscillations 14 min · Video

Gain and Phase Lag

  1. Definition of Stability Video
  2. Stability of Constant Coefficient Second Order DE's Video
  3. Gain and Phase Lag 11 min · Video

Undetermined Coefficients

  1. Undetermined Coefficients 11 min · Video

Linear Operators, Linear Time Invariance

  1. Solving DE's with Exponential Input when p(a) = 0 Video

Pure Resonance

  1. Resonance in the Undamped System Video
  2. Reconciling the Resonant and Non-resonant Solutions Video
  3. Pure Resonance 12 min · Video

Frequency Response and Practical Resonance

  1. Exploration of the Amplitude and Phase: Second Order II Applet 12 min · Video
  2. Frequency Response 15 min · Video

Fourier Series: Basics

  1. Introduction to Fourier Transform Video
  2. Fourier Series for Functions with Period 2L Video
  3. Orthogonality Relations Video
  4. Computing Fourier Series 15 min · Video

Operations on Fourier Series

  1. Even and Odd Functions Video
  2. Convergence of Fourier Series Video
  3. Exploration of the Fourier Coefficient Applet 11 min · Video
  4. Manipulating Fourier Series 15 min · Video

ODE's with Periodic Input, Resonance

  1. Solving ODE's, Resonant Frequencies Video
  2. Hearing I Video
  3. Hearing II Video
  4. Linear ODE's with Periodic Input 12 min · Video

Step and Delta Functions: Integrals and Generalized Derivatives

  1. Step and Delta Functions: Integration and Generalized Derivatives 10 min · Video

Unit Step and Unit Impulse Response

  1. Unit Step and Impulse Response 14 min · Video

Convolution

  1. Example: f(t)*1 Video
  2. Example: Radioactive Dumping Video
  3. Exploration of the Convolution Accumulation Applet 7 min · Video
  4. Convolution and Green's Formula 10 min · Video

Laplace Transform: Basics

  1. Laplace Transform Video
  2. Notation and Linearity Video
  3. Start of Laplace Table Video
  4. Functions of Exponential Type Video
  5. Laplace Transform of Powers Video
  6. Laplace Transform: Basics 10 min · Video

Partial Fractions and Inverse Laplace Transform

  1. Heaviside Coverup Method Video
  2. Partial Fractions and Laplace Inverse 13 min · Video

Laplace Transform: Solving Initial Value Problems

  1. Laplace: Solving ODE's 12 min · Video

Transfer (System) and Weight Functions, Green's Formula

  1. Laplace Transform of Convolution Video

Poles, Amplitude Response, Connection to ERF

  1. Pole Diagrams 11 min · Video

Linear Systems

  1. Introduction to Systems Video
  2. Autonomous Systems: Solutions are Parametrized Curves Video
  3. Linear Systems of Equations 7 min · Video

Matrix Methods: Eigenvalues and Normal Modes

  1. Matrix Methods Introduction, First Example Video
  2. Matrix Systems: General Terminology Video
  3. Complex Eigenvalues: Example Video
  4. Three Chamber Mixing Example: Repeated Roots Video
  5. Linear Systems: Matrix Methods 8 min · Video
  6. Linear Systems: Complex Roots 12 min · Video

Qualitative Behavior: Phase Portraits

  1. Sketching Solutions Video
  2. More Sketching Solutions Video
  3. Trace-Determinant Diagram 19 min · Video
  4. Phase Portraits 13 min · Video

Matrix Exponentials

  1. Theory of Systems Video
  2. Fundamental Matrices Video
  3. The Matrix Exponential I Video
  4. The Matrix Exponential II Video
  5. Example: Solving a System with the Matrix Exponential Video
  6. Calculating the Exponential Matrix Video
  7. Variation of Parameters Video
  8. Matrix Exponentials 12 min · Video
  9. Decoupling Video

Nonlinear Systems

  1. The Relation Between Nonlinear Systems and First-order ODE's Video

Linearization Near Critical Points

  1. The Nonlinear Pendulum Video
  2. Example: Predator-Prey and Volterra's Principle Video
  3. Linearization 16 min · Video

Limitations of the Linear: Limit Cycles and Chaos

  1. Limit Cycles Video
  2. Existence of Limit Cycles Video
  3. Non-existence of Limit Cycles Video
  4. A Story about Limit Cycles Video

Syllabus and materials

At MIT OpenCourseWare, as the course publishes them.

Credits and licence

Institution
Massachusetts Institute of Technology
Department
Mathematics
Taught by
Prof. Arthur Mattuck, Prof. Haynes Miller, Dr. Jeremy Orloff, Dr. John Lewis
Term
Fall 2011
Level
Undergraduate
Language
English
Subjects
Mathematics
Source
MIT OpenCourseWare
Licence
CC BY-NC-SA 4.0, as MIT OpenCourseWare states it. Non-commercial use only. MIT OpenCourseWare's terms
Attribution
Prof. Arthur Mattuck, Prof. Haynes Miller, Dr. Jeremy Orloff, Dr. John Lewis. Differential Equations. Fall 2011. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu/. License: Creative Commons BY-NC-SA.
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