MA 226 Section B Fall 2014

Welcome to the MA 226 Section B Homepage

Here is the class syllabus.

Unit 1 – First Order Differential Equations

  • Description – In this first unit we cover sections 1.1-1.9 of Chapter 1 where we introduce first
    order differential equations. We learn key words to classify these equations as linear or
    nonlinear, autonomous or non-autonomous, and homogeneous or non-homogeneous. We
    introduce three methods: separation of variables, method of undetermined coefficients, and
    integrating factors(for finding analytic solutions to some specific types of equations). We
    introduce a simple scheme called Euler’s Method for finding numerical solutions of differential
    equations. These topics will feel very natural to the students who have completed courses in
    differential and integral calculus and in some ways they are just an extension of those
    classes. What is new and exciting in this chapter is the introduction to a more qualitative
    understanding of the nature of first order differential equations. Our search for the
    asymptotic (long term) behavior of the solutions leads to a discussion of several new topics
    including : slope fields and phase lines, Existence and Uniqueness theorems, the connection
    between homogenous and non-homogenous equations, and the phenomenon of bifurcation.
  • Lecture Notes
  • Homework
  • Quizzes
  • Exam 1 – Tuesday September 30, 8-9:30  CAS314

Unit 2 -First Order Systems of Differential Equations and Linear Systems of Differential Equations

  • Description- In this second unit we cover sections 2.1-2.6 of Chapter two and sections 3.1-3.7
    of Chapter 3. Systems of differential equations describe the phenomenon of two related
    quantities that are changing in time. The predator prey model from section 1.1 is an example
    of the type of systems we will consider. In Chapter 2 we are introduced to vector notation
    which allows us to express a system as a first order homogeneous equation using vectors.
    In this form it is easy to visualize the behavior of systems by visualizing the behavior of the
    associated vector field. Also we can easily extend Euler’s Method and the Existence and
    Uniqueness theorems for systems expressed in a vector form. In Chapter 3 we consider the
    behavior of a very fundamental class of systems known as linear systems. Using matrix
    notation along with eigenvectors and eigenvalues we will explore the analytic solutions and
    the qualitative behavior of all possible 2-dimensional linear systems.
  • Lecture Notes
  • Homework
  • Quizzes
  • Exam 2 – Tuesday November 11, 8-9:30  CAS314

Unit 3 – Forcing and Resonance, Nonlinear Systems, and Laplace Transforms

  • Description-In this third unit we begin by building upon our understanding of the damped
    harmonic oscillator. Previously, the fate of the damped harmonic oscillator was to tend to the
    equilibrium rest position after an initial disturbance. In this unit we introduce the idea of a
    forcing function which can continue to excite the oscillator over time. We study the effects
    of three types of forcing functions: exponential, polynomial, and periodic. In the case of
    periodic forcing we will see in section 4.3 the phenomena called resonance whereby a small
    periodic forcing function can cause infinite amplitude oscillations. In Chapter 5 we introduce
    the idea of linearization of a nonlinear system near its equilibrium points so we can better
    understand the behavior at the equilibrium points. Chapter 6 introduces a very new idea
    called a Laplace Transform. Laplace Transforms have the ability to transform a differential
    equation into an algebraic equation. We will begin by showing that this technique produces
    the same solutions to familiar equations. However, one of the benefits of using Laplace
    Transforms is that it allows us to work with forcing functions that are not differentiable such as
    the Dirac delta function. The Dirac delta function models the effect of giving our system an
    instantaneous jolt of energy. For example, hitting the mass on a spring with a hammer. It is a
    very important type of forcing function that our methods in Chapter 4 cannot address.
  • Lecture Notes
  • Homework
  • Quizzes
  • Project
  • Exam 3 , Tuesday December 9, 8-9:30 CAS 314

Final Exam – Thursday December 18, 9-11 CAS 314