Math 321 Differential Equations

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Math 321 Differential Equations

• Instructor David Housman

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Motivation

• The study of differential equations and the
  calculus originated with the work of Isaac Newton
  and Gottfried Wilhelm Leibnitz. The development
  of these theories is considered one of the major
  intellectual achievements of the seventeenth
  century. The theories were developed in order to
  explain the physical motion of objects. Since
  that time, differential equations have been used
  to explain and predict the course of chemical
  reactions, animal populations, radioactive decay,
  military arms races, economic growth, the spread
  of diseases and rumors, and more. The techniques
  of differential equations are standard tools for
  engineers, physical scientists, and applied
  mathematicians.

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Review

• What are numbers, equations, and functions?
• numbers express quantities (3, -4.247, 0, ?2, ?,
  3 2i)
• equations express relationships (x2 y2 25)
• functions express dependency (h(t) 3 t2 7t
  4)
• What is common among numbers, equations, and
  functions?
• functions are not always expressed symbolically
• they are all objects
• What does solve an equation mean?
• find numbers that when substituted for the
  variables result in a true statement (x 3, y
  -4)

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Review

• What is a measurement?
• number, precision, units, and context
• What is a derivative?
• f ?(a) is the slope of the line tangent to the
  graph of y f(x) at the point (a, f(a)).
• f ?(a) is the instantaneous rate of change of f
  at a.
• if f (t) is the odometer reading of a car at
  time t, then f ?(t) is the velocity of the
  car at time t.
• f ?(a) limit??0 ( f (a ?) - f (a) ) / ?
• there are also many rules for calculating
  derivatives of functions defined symbolically

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Mathematical Modeling
Real World
Math World
phenomenonproblem
model
resultsolution
understandingsolution
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An Example

• Consider the human population of the United
  States.
• In groups of 2 or 3, make a list of reasonable
  assumptions about how human populations change.
• Meanwhile, I will take role.

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Population Growth

• Critical Idea If you double the population, then
  there will be twice as many births and deaths.
• Assumption The rate of population change is
  proportional to the population.
• Notation Let P(t) be the human population of
  the United States (expressed in millions) at time
  t (expressed in years).
• Model P ?(t) k P and P(t0) P0 where k,
  t0, and P0 are constant parameters.
• Solution Use graphic, analytic (guess and
  separation), and numeric techniques.