18.03 Differential Equations, Spring 2020

Welcome to the 18.03 master website! Here are some useful links/information:

• For changes to the course due to COVID-19, see below.
• Piazza, a forum where the students can ask questions about the material
• Course information sheet, read it carefully! (Updated March 31 due to COVID-19.)
• At the end of the course information sheet is important information on special accommodations (disability, religious, athletic etc.) which have to be requested far in advance!
• Learning Modules (Stellar): course announcements, materials, homework part B assignments
• MITx: homework part A, optional reading materials. Click here for an FAQ on signing in/submitting homework.
• Gradescope: homework part B submission (click here), exam grading (click here)
• Math Learning Center and Talented Scholars Resource Room, including tutoring
• See below for   instructors and office hours,   calendar,   exam information,   lecture notes,   and   syllabus

COVID-19 information

• Videos of recorded lectures are hosted on MIT OVS and available both under Lecture Notes (links to specific topics) and under Syllabus (whole videos). If the videos do not play well, try adjusting the bitrate. For Dropbox links to download lecture videos, click here. There are also references to MITx sections and videos (that could be used as supplementary material, or as backup in case OVS videos take too long to load).
• Recitations and office hours are happening via Zoom, click here for the links.
• MT2, MT3, Final Exam are cancelled, there is one more HW to account for this.
• Students are encouraged to post questions about the material on Piazza.
• For your entertainment, here is one introduction to (a simplified form of) the SEIR model of spread of infectious disease that could be used for COVID-19. It is discussed in §8.1 of the lecture notes.

Instructors and office hours

Any 18.03 student is welcome to come to any 18.03 recitation and office hour. Highly recommended! Click here for the Zoom links. All times are in US Eastern time zone. Office hours are subject to change, new times announced soon.

Calendar

Click here to add the class calendar to your Google Calendar

Exam information

The materials below are hosted on Stellar (they can be found under Materials/Exams). If the direct link from this website does not work, make sure you are signed in on Stellar.

• MT1: Friday February 28, 1:05–1:55 PM
  • Students with last names from A to Th will write the exam in 50-340 (top floor of Walker memorial). Students with last names from To to Z will write the exam in 26-100.
  • Covers everything up to and including §4.5.4. Sections 4.5.5–4.5.6 will not be on the exam.
  • Review session: Wednesday February 26, 7–9 PM in room 2-190
  • Review problems, solutions
  • First page of the exam containing important information, please read it before the exam
  • Exam solutions
  • Regrade requests are open on Gradescope until end of day next Thursday (strictly speaking, until next Friday March 6 at 4 AM).

Lecture notes and videos

• §1. Overview. Ordinary differential equations (ODEs): modeling, general solutions, initial value problem, qualitative behavior, illustrated on a simple example. [MITx: 1.1]
  • §1.1. Taxonomy of ODEs (notes in §1). Order, linear (in)homogeneous equations, constant/variable coefficients. [MITx: 1.1.5,1.1.6]
• §2. First order ODEs. Standard form of a first order ODE.
  • §2.1. Separable equations. Finding the general solution. Solving the initial value problem. [MITx: 1.2.2]
  • §2.2. Linear first order ODEs. Standard linear form. Homogeneous linear equations. Inhomogeneous equations: variation of parameters. Example: salting the tank. Steps for modeling. [MITx: 1.2.3,1.2.5,1.2.6,1.1.10,1.2.7]
• §3. Complex numbers.
  • §3.1. Algebraic properties of complex numbers. Definition of complex numbers, graphical representation, real and imaginary parts. Algebraic operations: addition, multiplication, conjugate, absolute value, division. Solving polynomial equations: quadratic equations, fundamental theorem of algebra, factorization. [MITx: 1.3.1–1.3.6, 1.4.5]
  • §3.2. The complex exponential. Definition of e^ib and e^a+ib via trigonometric functions using Euler's formula. The multiplicativity property. Differentiating complex exponentials. Polar form of a complex number, taking roots. [MITx: 1.3.7–1.3.12, 1.4.2, 1.4.6–1.4.8]
• §4. Linear ODEs with constant coefficients.
  • §4.1. Harmonic oscillator and superposition principle. Modeling a spring: harmonic oscillator. Linear combinations and linear independence of functions. General principles for linear homogeneous equations: superposition principle (linear combinations of solutions are again solutions), the form of a general solution. Initial value problem. [MITx: 1.5.2–1.5.4]
  • §4.2. General solution for 2nd order homogeneous equations. Characteristic equation and exponential substitution. The three cases: distinct real roots, double root, complex roots. Example: damped harmonic oscillator (dashpot) with the three cases: overdamping, critical damping, underdamping. [MITx: 1.5.5–1.5.11, 1.6.3, 1.7]
  • §4.3. Sinusoidal functions. Converting between the linear combination form, the complex form, and the amplitude-phase form. Graphing sinusoidal functions. [MITx: 1.6.4–1.6.12, 3.2.5–3.2.8, 3.2.10, 3.2.12]
  • §4.4. Higher order equations. The general solution. Existence and uniqueness for the initial value problem. [MITx: 1.8]
  • §4.5. Inhomogeneous equations. Operator notation. Inhomogeneous superposition principle. Exponential response formula (ERF). Complex replacement for sinusoidal inputs. ERF and the amplitude-phase form: complex gain, amplitude gain, and phase lag. Generalized ERF. [MITx: 1.9, 1.10.2–1.10.10]
  • §4.6. Electrical circuits, resonance, and stability. Modeling RLC circuits by damped harmonic oscillator. Amplitude gain for a damped harmonic oscillator. Exciting a harmonic oscillator, resonance phenomenon. Exciting at the resonant frequency: application of generalized ERF and complex replacement. Stability: definition, steady state/transient decomposition of solutions, tests for stability. For graphs of amplitude gain, click here. [MITx: 1.10.11–1.10.15, 1.11]
• §5. Linear algebra I.
  • §5.1. Vectors. Definition, the space Rⁿ, graphical representation of vectors in R² and R³. Basic operations: addition, multiplication by scalars, linear combination. Linear (in)dependence, span, basis. Coordinates of a vector in a basis. Basic properties of bases in Rⁿ. Canonical basis of Rⁿ. [MITx: 1.8.3–1.8.7, 2.1.3, 3.1.6–3.1.7]
  • §5.2. Matrices. Definition. Matrix as a linear map on Rⁿ. Example: matrix of a rotation. Identity matrix. Basic operations: addition, matrix multiplication. Null space. Determinant and trace of a 2×2 matrix. Eigenvalues and eigenvectors of 2×2 matrices, diagonalizability. [MITx: 2.1.5, 2.2.5–2.2.10]
  • §5.3. Vector spaces. Informal definition. Examples: Rⁿ; the space of functions. Linear combination, linear independence, span, basis. Finite dimensional vector spaces. Linear maps on vector spaces, null space, subspaces. [MITx: 1.8.4]
    Videos:    ■ vector spaces    ■ linear combinations, linear independence, basis, coordinates    ■ dimension    ■ linear maps, differential operators    ■ null space, subspaces, solutions to linear homogeneous ODEs    ■ more on vector spaces, basis, and coordinates (optional)
• §6. 2×2 systems of linear ODEs.
  • §6.1. Introduction to systems of ODEs. Linear homogeneous 2×2 systems with constant coefficients. The vector form. Converting a second order ODE into a first order system; the companion matrix. Example: fluid flow in two connected tanks. [MITx: 2.1]
    Videos:    ■ introduction to systems of ODEs    ■ some motivation with a partially true story about Newton    ■ vector notation for systems    ■ companion system of a 2nd order ODE    ■ example: two connected tanks
  • §6.2. Solving 2×2 systems using linear algebra. Writing the solution using eigenvectors when the matrix is diagonalizable. Initial value problem. Dealing with complex eigenvalues. [MITx: 2.2.3–2.2.4, 2.2.11–2.2.15]
    Videos:    ■ general solution using eigenvalues and eigenvectors    ■ general solution in case of complex eigenvalues    ■ example with real eigenvalues    ■ example with complex eigenvalues
  • §6.3. Phase portraits. Trajectories of solutions in the phase plane. Trace-determinant plane. Phase portraits case by case: source nodes, sink nodes, saddles, centers, spiral sources, spiral sinks, star sources, star sinks. [MITx: 2.3.3–2.3.14, 2.4.2–2.4.11] See MITx for nice pictures of phase portraits.
    Videos:    ■ trajectories of solutions    ■ phase portraits and physical morivation    ■ trace-determinant plane    ■ general comments    ■ 2 real eigenvalues: plotting special trajectories    ■ source nodes    ■ sink nodes    ■ saddles    ■ more saddles and an example    ■ another example of saddle and physical motivation    ■ sources, sinks, and saddles on the tr-det plane    ■ complex eigenvalues: general comments    ■ centers    ■ spiral sources and sinks    ■ centers and spiral sources/sinks on the tr-det plane    ■ double eigenvalue: star sources and sinks    ■ review and strategy
    MITx videos:    ■ sketching a solution    ■ sink and source nodes    ■ saddles    ■ complex eigenvalues    ■ spiral sources and sinks    ■ worked example of phase portraits
  • §6.4. Stability. Stability and semistability. Criteria for (semi)stability of 2×2 systems in terms of the eigenvalues and in terms of trace and determinant. [MITx: 2.4.12, 2.4.14]
    Videos:    ■ stability in terms of eigenvalues    ■ examples of stability and physical motivation    ■ stability on the trace-determinant plane
    MITx videos:    ■ phase portraits and stability on the tr-det plane
• §7. Linear algebra II.
  • §7.1. Systems of linear equations and elimination. Systems of linear equations and their vector form Ax=b. Augmented matrix. Row operations. Row echelon form (REF) and solving systems in REF by back substitution; dependent and free variables. Gaussian elimination to convert any matrix to REF. Reduced row echelon form (RREF). [MITx: 3.1]
    Videos:    ■ vector notation for systems of linear equations    ■ augmented matrix, row operations    ■ row echelon form    ■ strategy for solving linear systems    ■ solving systems in REF, dependent and free variables    ■ Gaussian elimination    ■ RREF
    MITx videos:    ■ worked example
  • §7.2. Null space, column space, rank, invertibility. Subspaces of Rⁿ. Null space of a matrix, finding its basis and dimension. Checking a set of vectors for linear independence. Column space of a matrix, finding its basis and dimension. The rank-nullity theorem. Invertible and singular square matrices, finding the inverse of a matrix. [MITx: 3.2, 3.3.2–3.3.12, 3.3.16]
    Videos:    ■ general discussion    ■ null space    ■ applications of null space; linear independence    ■ column space    ■ rank-nullity theorem    ■ existence and uniqueness for linear systems    ■ invertible matrices    ■ criteria for invertibility    ■ the inverse of a matrix
    MITx videos:    ■ introduction to vector spaces and subspaces    ■ example of nullspace    ■ worked examples    ■ linear dependence and independence    ■ linear independence and null space    ■ basis and dimension    ■ column space    ■ more worked examples    ■ geometry of solutions to linear equations    ■ inverse matrices
  • §7.3. Determinants, eigenvalues, and eigenvectors. Computing determinants. Determinants and invertibility. Eigenvalues, eigenvectors, and eigenspaces of a matrix. Diagonalizability and diagonalization. Eigenvalues, eigenvectors, and diagonalizability of symmetric matrices. [MITx: 3.3.13–3.3.15, 3.4]
    Videos:    ■ determinants    ■ eigenvalues, eigenvectors, eigenspaces, characteristic polynomial    ■ example of eigenvalue/eigenvector computation    ■ diagonalization    ■ example of diagonalization
    MITx videos:    ■ introduction to eigenvectors    ■ eigenvalues of the rotation matrix    ■ worked example of eigenvalues    ■ example of a non-diagonalizable matrix    ■ diagonalization
• §8. n×n systems of linear ODEs.
  • §8.1. Basics and examples. General form of an n×n system. First order companion systems of higher order ODEs and of higher order systems. Examples: the SEIR model of spread of infectious disease; a coupled spring system. Click here for an application of the SEIR model to COVID-19. [MITx: 3.5.10, 3.5.12–3.5.13]
    Videos:    ■ introduction and the SEIR model    ■ the linear SEIR model    ■ linear SEIR continued    ■ companion systems    ■ example: 3 coupled springs
    MITx videos:    ■ example: 3 connected tanks    ■ example: tuned mass damper
  • §8.2. Solving n×n systems. General solution for the case of diagonalizable A using eigenvalues and eigenvectors. Examples: the SEIR model; a coupled spring system. [MITx: 3.5.3, 3.5.14–3.5.16]
    Videos:    ■ finding the general solution    ■ general solution for linear SEIR    ■ 3 coupled springs: setup    ■ 3 coupled springs: eigenvalues and eigenvectors    ■ 3 coupled springs: general solution and dance
    MITx videos:    ■ solving the 3 tanks    ■ fundamental matrix    ■ fundamental matrix and the general solution    ■ experiment modeled by the 3 coupled spring system: carts on air tracks
  • §8.3. Exponential of a matrix and inhomogeneous systems. Matrix exponential, basic properties. Computing the exponential of a diagonalizable matrix. Matrix exponential and the general solution to a system of ODEs; initial value problems. Solving inhomogeneous systems by variation of parameters. [MITx: 3.6.13–3.6.16]
    Videos:    ■ motivation    ■ definition of matrix exponential    ■ basic properties of matrix exponential    ■ review of properties    ■ computing matrix exponential    ■ general solution to systems of ODEs    ■ interpretation of matrix exponential    ■ inhomogeneous systems of ODEs
    MITx videos:    ■ matrix exponential and fundamental matrix    ■ worked out example    ■ basic properties    ■ another worked out example    ■ variation of parameters
• §9. Boundary value problems.
  • §9.1. Basic properties. Definition of a boundary value problem (BVP) for a second order linear homogeneous ODE. Examples. Well-posed and ill-posed BVPs. [MITx: 4.4.3–4.4.5]
    Videos:    ■ introduction and motivation    ■ comparison with initial value problems    ■ solving BVPs, well-posedness    ■ examples
    MITx videos:    ■ BVP vs IVP
  • §9.2. Eigenvalues and eigenfunctions of differential operators. General setup for eigenvalues of second order differential operators. Eigenvalues and eigenfunctions of D² on an interval with Dirichlet and Neumann boundary conditions. [MITx: 4.4.6]
    Videos:    ■ introduction and motivation    ■ the Dirichlet case: introduction    ■ Dirichlet eigenvalues are nonpositive    ■ computing Dirichlet eigenvalues and eigenfunctions    ■ plots of Dirichlet eigenfunctions    ■ Neumann eigenvalues and eigenfunctions
• §10. Partial differential equations (PDEs) and Fourier series.
  • §10.1. The heat equation. Modeling heat distribution of a rod. Initial-boundary value problem with Dirichlet or Neumann boundary conditions. Special solutions arising from eigenfunctions. The general solution as a linear combination of special solutions. Solving the initial-boundary value problem; Fourier series. [MITx: 4.5]
    Videos:    ■ introduction    ■ modeling    ■ initial-boundary value problems    ■ Dirichlet case: special solutions    ■ Dirichlet case: general solution    ■ Dirichlet case: solving initial-boundary value problem; sine Fourier series    ■ Dirichlet case: example    ■ Neumann boundary conditions; cosine Fourier series
    MITx videos:    ■ deriving the heat equation    ■ general solution    ■ solving the initial-boundary value problem
  • §10.2. The wave equation. Modeling a vibrating string. General solution with Dirichlet boundary conditions. A musical interpretation. Solving the initial-boundary value problem using Fourier series. [MITx: 4.6]
    Videos:    ■ introduction    ■ modeling a vibrating string    ■ general solution    ■ musical interpretation    ■ solving the initial-boundary value problem
    MITx videos:    ■ modeling    ■ modeling continued    ■ general solution    ■ solving the initial-boundary value problem
  • §10.3. Theory of Fourier series. Inner product and orthogonality of functions. Finding the coefficients in sine and cosine Fourier series. Example: the function f(x)=1 (square wave). Periodic Fourier series. For a graph of sine Fourier series and solution to the heat equation for the square wave, click here. [MITx: 4.1, 4.2.2–4.2.9]
    Videos:   ■ introduction   ■ inner product on functions   ■ orthogonality of sines   ■ computing Fourier coefficients   ■ example: square wave
    MITx videos:   ■ motivation   ■ orthogonality   ■ proof of orthogonality   ■ Fourier coefficient formulas   ■ Fourier series of the square wave   ■ uniqueness of Fourier series   ■ even/odd Fourier series   ■ even/odd simplified formulas   ■ worked example of an odd periodic function   ■ convergence at jump discontinuities   ■ periodic extensions
• §11. Nonlinear ODEs: qualitative and numerical methods.
  • §11.1. Autonomous systems. Definition. Phase portraits. Critical points, linearization, stability. Example: physical pendulum. [MITx: 5.2.2–5.2.8, 5.4]
    Videos:   ■ introduction   ■ autonomous systems, phase portraits   ■ the physical pendulum   ■ critical points   ■ linearization around a critical point   ■ linearization for the physical pendulum   ■ phase portrait for the physical pendulum
    MITx videos:   ■ introduction   ■ example of critical points   ■ physical pendulum   ■ physical pendulum: companion system   ■ physical pendulum: critical points   ■ physical pendulum: linearization at the bottom   ■ physical pendulum: linearization   ■ physical pendulum: phase portrait
  • §11.2. Numerical methods. Euler's method, error estimate. A numerical demonstration of the SEIR model with and without social distancing. Click here for Matlab codes simulating the SEIR model (ModelSEIR.m is the main file). Here is another one written in Julia (with error bars). [MITx: 5.3]
    Videos:   ■ Euler's method   ■ numerical simulation of the SEIR model
    MITx videos:   ■ Euler's method

Syllabus and videos

Please see below for what sections of lecture notes are covered, as well as homework assignments due. The schedule for future lectures is subject to change. To access the homework assignment links below, you must be signed in on MITx (part A) or Learning Modules (part B). Part B can be found on Learning Modules under Gradebook/Assignments. The videos are accessible to registered students only.