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Math 2280 Details

Prerequisites: Math 2270 and its prerequisites are the prerequisite for Math 2280. Although it's not a formal prerequisite, Math 2280 students would also benfit from the multivariable calculus in either 1260, 1280, or 2210; they need an understanding of curves and tangent vectors to understand the geometric meaning of solutions to systems of differential equations, and they should understand partial derivatives and the chain rule to understand linearization (especially near equilibria of non-linear systems), and to make sense of partial differential equations.

Text: Differential Equations and Boundary Value Problems: Computing and Modeling, fourth edition, by by C.H. Edwards and D.E. Penney; ISBN=9780131561076. Essential course material is not much changed from previous editions, but problem sections are not necessarily the same.

Math 2280 course outline: The semester begins with first order differential equations: their origins, geometric meaning (slope fields), analytic and numerical solutions. The logistic equation and various velocity and acceleration models are studied closely. The next topic is linear DE's of higher order, with the principal application being mechanical vibrations (friction, forced oscillations, resonance).

At this point we show how various scientific models of dynamical systems lead to first order systems of differential equations. I explain the connection between the discrete dynamical systems from Math 2270, and the continuous version here. Use the linear algebra from Math 2270 to develop the theory and techniques for solving first order linear systems of differential equations.

With the linear theory well in-hand, the concepts of the phase plane, stability, periodic orbits and dynamical-system chaos are introduced with various ecological and mechanical models.

The study of ordinary differential equations concludes with an introduction to the Laplace transform.

The final portion of Math 2280 is an introduction to the classical partial differential equations: the heat, wave and Laplace equations, and to the use of Fourier series and separation of variable ideas to solve these equations in special cases. Time permitting, one can also introduce the Fourier transform.

2280 suggested lectures: The following estimates add up to 50 lectures. In the typical fall (spring) term there are 58 (resp 57) MTWF class meetings, leaving some time for Maple labs, reviews, and exams. If you find this schedule too tight, you might omit 3.7 and 9.7, for example. Do not short-change chapters 1-6, and try to spend enough time on chapter 9 so that students get some feel for partial differential equations. (In recent years I have not always been able to follow this last recommendation. But I have always managed to revisit the forced oscillation problem using Fourier series, i.e. thru section 9.4)
  • Chapter 1: First-order differential equations - 5 lectures, e.g. one each for 1.1-1.5
  • Chapter 2: Mathematical Models and Numerical Methods - 5 lectures, e.g. one each for 2.1, 2.2; two lectures for 2.3; one survey lecture for 2.4-2.6.
  • Chapter 3: Linear Equations of Higher Order - 8 lectures, e.g. one for 3.1-3.2; one each for 3.3, 3.4, 3.5; two for 3.6; one for 3.7.
  • Chapter 4: Introduction to Systems of Differential Equations - 2 lectures, e.g. one each for 4.1 and 4.3.
  • Chapter 5: Linear Systems of Differential Equations - 7 lectures, e.g. one for 5.1-5.2; one each for 5.3, 5.4; two for 5.5; one for 5.6.
  • Chapter 6: Nonlinear Systems and Phenomena - 6 lectures, e.g. two each for 6.1-6.2 and for 6.3-6.4; one for 6.5.
  • Chapter 7: Laplace Transform}- 5 lectures, e.g. one each for 7.1, 7.2, 7.3, 7.4, 7.6.
  • Chapter 9: Fourier Series Methods - 12 lectures, e.g. two for 9.1, one for 9.2, two for 9.3, one for 9.4, two each for 9.5, 9.6, 9.7.
Computer projects: Aim for at least 3 substantive projects, in addition to regular homework which has computational aspects. Possible topics for 2280 projects include:
  • Slope fields, Euler's method for 1st-order differential equations (Chapters 1-2
  • Newton's law of cooling for a house (Chapter 1)
  • Modeling populations with the logistic equations (Chapter 2)
  • Modeling springs: damping, forced oscillations, resonance and approximate resonance (Chapter 3)
  • Earthquakes and multi-story building vibrations (Chapter 5)
  • "pplane" investigation of non-linear phase portraits (Chapter 6)
  • Numerical methods for systems, chaos in Duffing's spring equation (Chapter 6)
  • Fourier series (Chapter 9)
  • Fourier series solutions to the heat and wave equations (Chapter 9)
Recent course home pages: Recent instructors of Math 2280 here at Utah include Jing Tao, myself, Grant Gustafson and Dylan Zwick. We can be excellent resources if you have questions about teaching this class. Here are links to our course home pages, which likely include syllabus, exam, project and homework details. These may be very helpful as you construct and teach your own course: