Differential Equations (MIT 18.03) - Worked examples 2011
Lydia Bourouiba and David Shirokoff

@article{,
title= {Differential Equations (MIT 18.03) - Worked examples 2011},
journal= {},
author= {Lydia Bourouiba and David Shirokoff},
year= {2011},
url= {http://ocw.mit.edu/18-03SCF11},
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/18-03SCF11

---

**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) - Supplementary notes 2010](https://academictorrents.com/details/1d3ced525d5c5bfbcaa71e176c371c373e4122dc)
* [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
    * Recitation 110708_L1: Separable Equations
    * Recitation 110708_D1: Direction Fields
  * Lecture 2: Euler's Numerical Method for y'=f(x,y) and its Generalizations
    * Recitation 110708_D2: Euler's Method
  * Lecture 3: Solving First-order Linear ODE's (Ordinary Differential Equations); Steady-state and Transient Solutions
    * Recitation 110708_D3: Linear Equations
    * Recitation 110708_L2: Solutions of First Order Linear Equations
  * Lecture 4: First-order Substitution Methods: Bernoulli and Homogeneous ODE's
  * Lecture 5: First-order Autonomous ODE's: Qualitative Methods, Aplications
  * Lecture 6: Complex Numbers and Complex Exponentials
    * Recitation 110726_L1: Complex Numbers and Euler's Formula
  * 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
    * Recitation 110720_D1: Sinusoidal Functions
    * Recitation 110720_D2: First Order Constant Coefficient Linear ODE's
    * Recitation 110726_L2: Sinusoidal Inputs
    * Recitation 110720_D3: Autonomous Equations and Phase Lines


* Unit II: Second Order Constant Coefficient Linear Equations
  * Lecture 9: Solving Second-order Linear ODE's with Constant Coefficients: The Three Cases
  * Lecture 10: Continuation: Complex Characteristic Roots; Undamped and Damped Oscillations
    * Recitation 110720_D4: Homogeneous Constant Coefficient Equations: Real Roots
    * Recitation 110726_L5: Homogeneous Constant Coefficient Equations: Any Roots
    * Recitation 110804_L4: Damped Harmonic Oscillators
  * Lecture 11: Theory of General Second-order Linear Homogeneous ODE's: Superposition, Uniqueness, Wronskians
  * 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
    * Recitation 110720_D5: Forced Oscillations
    * Recitation 110728_D3: Gain and Phase Lag
    * Recitation 110720_D6: Undetermined Coefficients
  * Lecture 14: Interpretation of the Exceptional Case: Resonance
    * Recitation 110802_L5: Pure Resonance
    * Recitation 110726_D3: Frequency Response


* Unit III: Fourier Series and Laplace Transform
  * Lecture 15: Introduction to Fourier Series; Basic Formulas for Period 2π.
  * Lecture 16: Continuation: More General Periods; Even and Odd Functions; Periodic Extension
    * Recitation 110726_D4: Computing Fourier Series
    * Recitation 110727_D3: Manipulating Fourier Series
  * Lecture 17: Finding Particular Solutions via Fourier Series; Resonant Terms; Hearing Musical Sounds
    * Recitation 110727_D4: Linear ODE's with Periodic Input
    * Recitation 110728_L2: Step and Delta Functions: Integration and Generalized Derivatives
    * Recitation 110728_L7: Unit Step and Impulse Response
  * Lecture 19: Introduction to the Laplace Transform; Basic Formulas
  * Lecture 20: Derivative Formulas; Using the Laplace Transform to Solve Linear ODE's
  * Lecture 21: Convolution Formula: Proof, Connection with Laplace Transform, Application to Physical Problems
    * Recitation 110728_D4: Convolution and Green's Formula
    * Recitation 110728_L1: Laplace Transform: Basics
    * Recitation 110728_D5: Partial Fractions and Laplace Inverse
    * Recitation 110728_D6: Laplace: Solving ODE's
    * Recitation 110802_L1: Pole Diagrams
  * Lecture 22: Using Laplace Transform to Solve ODE's with Discontinuous Inputs
  * Lecture 23: Use with Impulse Inputs; Dirac Delta Function, Weight and Transfer Functions


* Unit IV: First-order Systems
  * Lecture 24: Introduction to First-order Systems of ODE's; Solution by Elimination, Geometric Interpretation of a System.
    * Recitation 110802_L2: Linear Systems of 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
    * Recitation 110728_L8: Linear Systems: Matrix Methods
    * Recitation 110802_L3: Linear Systems: Complex Roots
  * Lecture 27: Sketching Solutions of 2x2 Homogeneous Linear System with Constant Coefficients
    * Recitation 110804_L1: Trace-Determinant Diagram
    * Recitation 110804_L2: Phase Portraits
  * 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
    * Recitation 110802_L4: Matrix Exponentials
  * 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
    * Recitation 110804_L3: Linearization
  * 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

---

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