Differential Equations (MIT 18.03) - Supplementary notes 2010
Haynes Miller

@article{,
title= {Differential Equations (MIT 18.03) - Supplementary notes 2010},
journal= {},
author= {Haynes Miller},
year= {2010},
url= {http://ocw.mit.edu/18-03S06},
abstract= {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 linear differential equations and their applications in science and engineering.

**Main course URL:** http://ocw.mit.edu/18-03S06

---

**Prerequisites:**
* Multivariable Calculus (MIT 18.02) - [Video lectures](https://academictorrents.com/details/81d27851df2879ed1307c8d60980d2b8af19264b) and [Worked examples](https://academictorrents.com/details/28a30bdd9a6859622e87e04ffb9eb4a9810bdf77)

**Related materials:**
* [Differential Equations (MIT 18.03) - Video lectures 2003](https://academictorrents.com/details/1bd4f7e3b3c0505869e5c1b54e98af54b125d05e)
* [Differential Equations (MIT 18.03) - Worked examples 2011](https://academictorrents.com/edit/6c3432a82526a1f25ca0edc4e9f09254da66aeb2)
* [Notes on Diffy Qs: Differential Equations for Engineers - Textbook, edition 6.11, 2026](https://academictorrents.com/details/7bacfc3ee249d6506c16ed1df2bbb19d825eb665)

---
**Table of contents:**

* Unit I: First Order Differential Equations
  * Lecture 1: The Geometrical View of y'=f(x,y): Direction Fields, Integral Curves
    * Chapter 1: Notation and Language
  * Lecture 2: Euler's Numerical Method for y'=f(x,y) and its Generalizations
  * Lecture 3: Solving First-order Linear ODE's (Ordinary Differential Equations); Steady-state and Transient Solutions
    * Chapter 2: Modeling by First Order Linear ODEs
    * Chapter 3: Solutions of First Order Linear ODEs
  * Lecture 4: First-order Substitution Methods: Bernoulli and Homogeneous ODE's
  * Lecture 5: First-order Autonomous ODE's: Qualitative Methods, Aplications
    * Appendix A: The Kermack-McKendrick Equation
  * Lecture 6: Complex Numbers and Complex Exponentials
    * Chapter 5: The Algebra of Complex Numbers
    * Chapter 6: The Complex Exponential
  * Lecture 7: First-order Linear with Constant Coefficients: Behavior of Solutions, Use of Complex Methods
  * Lecture 8: Continuation; Applications to Temperature, Mixing, RC-circuit, Decay, and Growth Models
    * Chapter 4: Sinusoidal Solutions


* Unit II: Second Order Constant Coefficient Linear Equations
  * Lecture 9: Solving Second-order Linear ODE's with Constant Coefficients: The Three Cases
    * Chapter 9: Normalization of Solutions
  * Lecture 10: Continuation: Complex Characteristic Roots; Undamped and Damped Oscillations
  * Lecture 11: Theory of General Second-order Linear Homogeneous ODE's: Superposition, Uniqueness, Wronskians
    * Chapter 15: The Wronskian
  * Lecture 12: Continuation: General Theory for Inhomogeneous ODE's. Stability Criteria for the Constant-coefficient ODEs
  * Lecture 13: Finding Particular Solutions to Inhomogeneous ODE's: Operator and Solution Formulas Involving Exponentials
    * Chapter 7: Beats
    * Chapter 10: Operators and the Exponential Response Formula
  * Lecture 14: Interpretation of the Exceptional Case: Resonance
    * Chapter 8: RLC Circuits
    * Chapter 11: Undetermined Coefficients
    * Chapter 12: Resonance and the Exponential Shift Law
    * Chapter 13: Natural Frequency and Damping Ratio
    * Chapter 14: Frequency Response
    * Appendix B: The Tacoma Narrows Bridge, Resonance vs Flutter


* Unit III: Fourier Series and Laplace Transform
  * Lecture 15: Introduction to Fourier Series; Basic Formulas for Period 2π.
    * Chapter 16: More on Fourier Series
  * Lecture 16: Continuation: More General Periods; Even and Odd Functions;  Periodic Extension
  * Lecture 17: Finding Particular Solutions via Fourier Series; Resonant Terms; Hearing Musical Sounds
  * Lecture 19: Introduction to the Laplace Transform; Basic Formulas
  * Lecture 20: Derivative Formulas; Using the Laplace Transform to Solve Linear ODE's
    * Chapter 20: Laplace Transform Technique: Cover-up
    * Chapter 21: The Laplace Transform and Generalized Functions
  * Lecture 21: Convolution Formula: Proof, Connection with Laplace Transform, Application to Physical Problems
    * Chapter 19: Convolution
  * Lecture 22: Using Laplace Transform to Solve ODE's with Discontinuous Inputs
    * Chapter 22: The Pole Diagram and the Laplace Transform
    * Chapter 23: Amplitude Response and the Pole Diagram
    * Chapter 24: The Laplace Transform and more General Systems
  * Lecture 23: Use with Impulse Inputs; Dirac Delta Function, Weight and Transfer Functions
    * Chapter 17: Impulses and Generalized Functions
    * Chapter 18: Impulse and Step Responses


* Unit IV: First-order Systems
  * Lecture 24: Introduction to First-order Systems of ODE's; Solution by Elimination, Geometric Interpretation of a System.
    * Chapter 25: First Order Systems and Second Order Equations
  * Lecture 25: Homogeneous Linear Systems with Constant Coefficients: Solution via Matrix Eigenvalues (Real and Distinct Case)
  * Lecture 26: Continuation: Repeated Real Eigenvalues, Complex Eigenvalues
  * Lecture 27: Sketching Solutions of 2x2 Homogeneous Linear System with Constant Coefficients
    * Chapter 26: Phase Portraits in Two Dimensions
  * Lecture 28: Matrix Methods for Inhomogeneous Systems: Theory, Fundamental Matrix, Variation of Parameters
  * Lecture 29: Matrix Exponentials; Application to Solving Systems
  * Lecture 30: Decoupling Linear Systems with Constant Coefficients
  * Lecture 31: Non-linear Autonomous Systems: Finding the Critical Points and Sketching Trajectories; the Non-linear Pendulum
  * Lecture 32: Limit Cycles: Existence and Non-existence Criteria
  * Lecture 33: Relation Between Non-linear Systems and First-order ODEs; Structural Stability of a System, Borderline Sketching Cases; Illustrations Using Volterra's Equation and Principle
    * Appendix B. The Tacoma Narrows Bridge: Resonance vs Flutter
    * Appendix C: Linearization The Phugoid Equation as Example
},
keywords= {Math, Maths, mathematics, calculus},
terms= {https://ocw.mit.edu/terms},
license= {https://creativecommons.org/licenses/by-nc-sa/4.0/},
superseded= {}
}