Ordinary Differential Equations and Linear Algebra

Fall semester 1998

  • Prerequisites
  • Course Outline
  • Grading
  • Tentative Daily Schedule
  • Texts:

    Differential Equations and Boundary Value Problems, Computing and Modeling, and Computer Projects supplement by C.H. Edwards Jr. and David E. Penney

    Analytical and Computational methods of Advanced Engineering Mathematics, chapter 5 supplement by Grant B. Gustafson and Calvin H. Wilcox

    Time: MWF 12:55-1:45 JTB 120

    Instructor: Nick Korevaar

    Web Page:http://www.math.

    Office: JWB 218

    Office Hours: M 2-2:50 p.m., T 12:30-2:30 p.m., W 2-2:50 p.m., F 8:20-9:00 a.m.

    Telephone: 581-7318




    Math 1210 and 1220 or the equivalent (first-year Calculus, with a very brief introduction to linear differential equations).This was Math 111-112-113 last year. In addition you are expected to be familiar with vectors, curves, velocity (tangent), and acceleration vectors from Physics 2210 or Math 2210, or their equivalents.

    Course Outline:

    This course is an introduction to differential equations, and how they are used to model problems arising in engineering and science. Linear algebra is introduced as a tool for analyzing systems of differential equations, as well as standard linear equations. Particular attention is paid throughout the course to qualitative behavior of solutions, stability, and numerical approaches. Computer projects will be assigned to enhance the material.

    We will cover most of chapters 1-8 in the Edwards-Penney text, as well as chapter 5 from the Gustafson-Wilcox text. If you intend eventually to take ``Partial Differential Equations for Engineers'', you should buy the entire Gustafson-Wilcox text since it is the book used in that course. Even if you do not plan on taking Math 3150 later, the Gustafson-Wilcox book provides another viewpoint to the material in Edwards-Penney and you may wish to purchase it as an additional text for this course. You are only required to purchase the chapter 5 supplement, however.

    The course begins with first order differential equations, a subject which you touched on in Calculus. Recall that a differential equation is an equation involving an unknown function and its derivatives, that such equations often arise in science, and that the order of a differential equation is defined to be the highest order derivative occurring in the equation. The goal is to understand the solution functions to differential equations since these functions will be describing the scientific phenomena which led to the differential equation in the first place. In chapters one and two of Edwards-Penney we learn analytic techniques for solving certain first order DE's, the geometric meaning of graphs of solutions, and the numerical techniques for approximating solutions which are motivated by this geometric interpretation. We will carefully study the logistic population growth model from mathematical biology and various velocity-acceleration models from physics.

    In chapter 3 we will study the theory of higher-order linear DE's, and focus on the second order ones which describe basic mechanical and electrical vibrations. You were introduced to these in Calculus as well, but we will treat them more completely now. We will study forced oscillations and resonance in this setting. Then in chapter 4 we will introduce systems of (several) differential equations (for several related unknown functions) and indicate how these arise naturally in dynamical systems in which it takes several functions to describe the complete behavior. We will study numerical methods for approximating solutions but will delay the analytic theory (chapters 5-6) until the latter part of the course.

    Our next topic will be the Laplace transform and its applications to the study of linear DE's, chapter 7. This ``magic'' transform takes differential equations and ``transforms'' them into algebraic equations. You will have to see it in action to appreciate it. You will see, for example, that this method gives a powerful way to study forced oscillations in the physically important cases that the forcing terms are step functions or impulse functions. Finally, in chapter 8, we will introduce our last technique for studying (single) differential equations, the method of power series. It is based on the idea of Taylor series, which you saw in Calculus.

    At this point in the course we will take a month-long digression to learn the fundamentals of linear algebra - a field of mathematics which we need to understand in order to talk meaningfully about the theory of higher order linear DE's and of systems of linear DE's. We will use the chapter 5 supplement from Gustafson-Wilcox. The chapter starts out with matrix equations and the Gauss-Jordan method of solution. When you see such equations in high-school algebra you might be thinking of intersecting lines in the plane, or intersecting planes in space, or ways to balance chemical reactions, but the need to understand generally how to solve such equations is pervasive in science. From this concrete beginning we study abstract vector spaces and linear operators. Not only is such abstract theory useful in studying linear maps (or differentiable maps) between Euclidean spaces, but it is the framework which allows us to understand solution spaces to systems of linear differential equations as well. Along the way we will discuss the subtopics of determinants, eigenvalues and eigenvectors.

    After the linear algebra digression we return to chapter 5 of Edwards-Penney and apply our theory to understand the solutions of systems of linear differential equations. We then study fundamental modes and resonance questions for forced oscillations in complicated mechanical systems like multi-story buildings. Throughout the course we will be showing how linear differential equations approximate (more accurate) nonlinear ones near points of equilibrium. In chapter 6 we will study global phenomena for nonlinear DE's, including phase-plane analysis and the notion of chaos.

    There will be four computer projects assigned during the semester, related to the classroom material. They will be written in the software package MAPLE. There is a Math Department Computer Lab at which you all automatically have accounts, and there are other labs around campus where Maple is also available, for example at the College of Engineering. There will tutoring center support for these projects (and for your other homework) as well. The Math Department Lab and Engineering Math Tutoring Center are both located in Building 129, between JWB and LCB.

    You will be permitted to work your solutions to the computer projects in different languages, such as Mathematica or Matlab, if you prefer these, although it will be easier to find help if you are using Maple. Notice that the text and the Computing Projects supplement support all three of these language however.



    There will be two midterms, a comprehensive final examination, and homework. Each midterm will count for 20% of your grade, homework will count for 30%, and the final exam will make up the remaining 30%. The book homework will be assigned daily and collected weekly, on Fridays. Maple project homework will generally be due one week after it is assigned. A grader will partially grade your assignments. The value of carefully working homework problems is that mathematics (like anything) must be practiced and experienced to be learned.

    It is the Math Department policy, and mine as well, to grant any withdrawal request until the University deadline of October 23.

    ADA statement:

    The American with Disabilities Act requires that reasonable accommodations be provided for students with physical, sensory, cognitive, systemic, learning, and psychiatric disabilities. Please contact me at the beginning of the semester to discuss any such accommodations for the course.

    Tentative Daily Schedule

    F 28 Aug1.1 & 1.2modeling and differential equations
    M31 Aug1.3slope fields and solution curves
    W2 Sept1.4, Maple intro.separable differential equations, begin tutorial
    F4 Sept1.5first order linear DE's
    M7 SeptnoneLabor Day
    W9 Sept2.1population model
    F11 Sept2.2equilibrium solutions and stability
    M14 Sept2.3acceleration-velocity models
    W 16 Sept 2.4 & 2.6 numerical techniques: Euler and Runge Kutta
    " " " " Maple I begin logistic equation, numerics in discretization
    F 18 Sept 3.1& 3.2 introduction and general theory of linear DE's
    M 21 Sept 3.3 homogeneous equations with constant coefficients
    W 23 Sept 3.4 mechanical vibrations
    F 25 Sept 3.5 nonhomogeneous equations
    M 28 Sept 3.6 forced oscillations and resonance
    W 30 Sept 3.7 electrical circuits
    F 2 Oct 4.1 first order systems of DE's
    M 5 Oct 4.3 numerical methods for systems
    W 7 Oct exam 1 1/4
    F 9 Oct none fall break day
    M 12 Oct 4.3 & 6.5 numerics for Duffing equation, Maple
    " "" " Maple II begin resonance in forced oscillations, chaos in Duffing
    W 14 Oct 7.1 Laplace transform intro.
    F 16 Oct 7.2 transforms of initial value problems
    M 19 Oct 7.3 translation and partial fractions
    W 21 Oct 7.4 derivatives, integrals and products of transforms
    F 23 Oct 7.6 impulses and delta functions
    M 26 Oct 8.1 introduction and review of power series
    W 28 Oct 8.2 series solutions near ordinary points
    F 30 Oct 5.1 G-W systems of linear equations
    M2 Nov5.2 G-WGaussian elimination
    W4 Nov5.3 G-Wvector spaces
    F6 Nov5.3 G-Wcontinued
    M9 Nov5.4 G-Wmatrices and matrix algebra
    W11 Nov5.4 G-Wcontinued
    " " " "Maple IIIbegin computer linear algebra
    F13 Nov5.5 G-Wfundamental theorem of linear algebra
    M16 Nov5.5 G-Wcontinued
    W18 Nov5.6 G-Wdeterminants and Cramer's rule
    F20 Nov5.6 G-Wcontinued
    M23 Novexam 27-8; 5.1-5.6 G-W
    W25 Nov5.7 G-Weigenvalues and eigenvectors
    F27 NovnoneThanksgiving
    M30 Nov5.7 G-Wcontinued
    W2 Dec5.1-5.2systems of DE's theory, eigenvector sol method
    F4 Dec5.3big mechanical vibrations
    " "" "Maple IVbegin vibrations in multistory buildings
    M7 Dec6.1-6.2phase plane analysis
    W9 Dec6.4nonlinear mechanical systems
    F11 Decallentire course
    Th17 DecFINAL EXAMentire course 12:30-2:30