Here is the class syllabus<\/a>.<\/p>\n Unit 1 – First Order Differential Equations<\/p>\n Unit 2 -First Order Systems of Differential Equations and Linear Systems of Differential Equations<\/p>\n Unit 3 \u2013 Forcing and Resonance, Nonlinear Systems, and Laplace Transforms<\/p>\n Final Exam – Thursday December 18, 9-11 CAS 314<\/p>\n <\/p>\n <\/p>\n <\/p>\n <\/p>\n","protected":false},"excerpt":{"rendered":" Welcome to the MA 226 Section B Homepage Here is the class syllabus. Unit 1 – First Order Differential Equations Description \u2013 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, […]<\/p>\n","protected":false},"author":1301,"featured_media":0,"parent":0,"menu_order":1,"comment_status":"closed","ping_status":"closed","template":"","meta":[],"_links":{"self":[{"href":"https:\/\/sites.bu.edu\/david-deutsch\/wp-json\/wp\/v2\/pages\/5"}],"collection":[{"href":"https:\/\/sites.bu.edu\/david-deutsch\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sites.bu.edu\/david-deutsch\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sites.bu.edu\/david-deutsch\/wp-json\/wp\/v2\/users\/1301"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.bu.edu\/david-deutsch\/wp-json\/wp\/v2\/comments?post=5"}],"version-history":[{"count":50,"href":"https:\/\/sites.bu.edu\/david-deutsch\/wp-json\/wp\/v2\/pages\/5\/revisions"}],"predecessor-version":[{"id":525,"href":"https:\/\/sites.bu.edu\/david-deutsch\/wp-json\/wp\/v2\/pages\/5\/revisions\/525"}],"wp:attachment":[{"href":"https:\/\/sites.bu.edu\/david-deutsch\/wp-json\/wp\/v2\/media?parent=5"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}\n
\norder differential equations. We learn key words to classify these equations as linear or
\nnonlinear, autonomous or non-autonomous, and homogeneous or non-homogeneous. We
\nintroduce three methods: separation of variables, method of undetermined coefficients, and
\nintegrating factors(for finding analytic solutions to some specific types of equations). We
\nintroduce a simple scheme called Euler\u2019s Method for finding numerical solutions of differential
\nequations. These topics will feel very natural to the students who have completed courses in
\ndifferential and integral calculus and in some ways they are just an extension of those
\nclasses. What is new and exciting in this chapter is the introduction to a more qualitative
\nunderstanding of the nature of first order differential equations. Our search for the
\nasymptotic (long term) behavior of the solutions leads to a discussion of several new topics
\nincluding : slope fields and phase lines, Existence and Uniqueness theorems, the connection
\nbetween homogenous and non-homogenous equations, and the phenomenon of bifurcation.<\/li>\n\n
\nof Chapter 3. Systems of differential equations describe the phenomenon of two related
\nquantities that are changing in time. The predator prey model from section 1.1 is an example
\nof the type of systems we will consider. In Chapter 2 we are introduced to vector notation
\nwhich allows us to express a system as a first order homogeneous equation using vectors.
\nIn this form it is easy to visualize the behavior of systems by visualizing the behavior of the
\nassociated vector field. Also we can easily extend Euler\u2019s Method and the Existence and
\nUniqueness theorems for systems expressed in a vector form. In Chapter 3 we consider the
\nbehavior of a very fundamental class of systems known as linear systems. Using matrix
\nnotation along with eigenvectors and eigenvalues we will explore the analytic solutions and
\nthe qualitative behavior of all possible 2-dimensional linear systems.<\/li>\n\n
\nharmonic oscillator. Previously, the fate of the damped harmonic oscillator was to tend to the
\nequilibrium rest position after an initial disturbance. In this unit we introduce the idea of a
\nforcing function which can continue to excite the oscillator over time. We study the effects
\nof three types of forcing functions: exponential, polynomial, and periodic. In the case of
\nperiodic forcing we will see in section 4.3 the phenomena called resonance whereby a small
\nperiodic forcing function can cause infinite amplitude oscillations. In Chapter 5 we introduce
\nthe idea of linearization of a nonlinear system near its equilibrium points so we can better
\nunderstand the behavior at the equilibrium points. Chapter 6 introduces a very new idea
\ncalled a Laplace Transform. Laplace Transforms have the ability to transform a differential
\nequation into an algebraic equation. We will begin by showing that this technique produces
\nthe same solutions to familiar equations. However, one of the benefits of using Laplace
\nTransforms is that it allows us to work with forcing functions that are not differentiable such as
\nthe Dirac delta function. The Dirac delta function models the effect of giving our system an
\ninstantaneous jolt of energy. For example, hitting the mass on a spring with a hammer. It is a
\nvery important type of forcing function that our methods in Chapter 4 cannot address.<\/li>\n\n