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2250-1 7:30am Lecture Record Week 3 S2012

Last Modified: January 29, 2012, 19:34 MST.    Today: October 19, 2017, 03:19 MDT.

Week 3, Jan 23 to 27: Sections 2.1, 2.2, 2.3, 2.4

23 Jan: Autonomous systems and applications section 2.1

Drill on 1.5 Problems
  There are two special methods for solving y'+py=q
     If p,q are constant then use the SUPERPOSITION METHOD
         y = y_p + y_h
         y_p = an equilibrium solution (set y'=0, solve for y)
         y_h = constant divided by the integrating factor
     If one of p or q depends on x, then use the STANDARD METHOD
         Replace the LHS, which is y'+p(x)y, by the integrating 
         factor quotient (Wy)'/W, where W=exp(int p(x)dx)) is the
         integrating factor.
         Cross-multiply by W to clear fractions. Then apply the
         method of quadrature. 

Slides: Linear integrating factor method (129.8 K, pdf, 03 Mar 2012)
Manuscript: Linear DE part I. Integrating Factor Method (152.7 K, pdf, 07 Aug 2009)
General Verhulst DE
    Solving y'=(a-by)y by a substitution
       Let u=y/(a-by).
       Then substitution into the DE gives u'=au
       Solve u'=au to get u=u0 exp(ax).
       Back-substitute u(x) into u=y/(a-by), then solve for y.
   Solving y'=(a-by)y by partial fractions
       Divide the DE by (a-by)y
       Apply the method of quadrature.
       Find the constants in the partial fractions on the left.
       Integrate to get the answer
                       a y0
       y(x) = --------------------------
              b y0 + (a - b y0) exp(-ax)
       where y0=y(0)=initial population size.
   

24 Jan: Autonomous Differential Equations and Phase Diagrams. Sections 2.1, 2.2

Lecture on 2.2:
  Theory of autonomous DE y'=f(y)
     Picard's theorem and non-crossing of solutions.
     Direction fields and translation of solutions
  Constructing Euler's threaded solution diagrams
     No direction field is needed to draw solution curves
       We throw out the threaded solution rule used in chapter 1,
       replace it by two rules from calculus and a theorem:
          1. If y'(x)>0, then y(x) increases.
          2. If y'(x)<0, then y(x) decreases.
          THEOREM. For y'=f(y), a threaded solution starting with
              y'(0)>0 must satisfy y'(x)>0 for x>0. A similar result
              holds for y'(0)<0.
     Definition: phase line diagram, phase diagram,
       Calculus tools: f'(x) pos/neg ==> increasing/decreasing
       DE tool: solutions don't cross
       Maple tools for production work.
  Stability theory of autonomous DE y'=f(y)
    Stability of equilibrium solutions.
    Stable and unstable classification of equilibrium solutions.
    funnel, spout, node,
  How to construct Phase line diagrams
  How to make a phase diagram graphic
    Inventing a graph window
    Invention of the grid points
    Using the phase line diagram to make the graphic
        calculus tools
        DE tools
    References for 2.1, 2.2, 2.3. Includes the rabbit problem, partial fraction examples, phase diagram illustrations.
    Slides: Autonomous DE (102.3 K, pdf, 03 Mar 2012)
    Manuscript: Verhulst logistic equation (115.5 K, pdf, 02 Oct 2009)
    Manuscript: Phase Line and Bifurcation Diagrams. Includes Stability, Funnel, Spout, and bifurcation (227.4 K, pdf, 07 Sep 2009)
    Text: Problem notes 2.1-8,16 (1.4 K, txt, 27 Jan 2012)
    Text: Problem notes 2.2-10,14 (1.8 K, txt, 28 Jan 2012)
    Text: Problem notes 2.3-10,20 (2.8 K, txt, 28 Jan 2012)
    Transparencies: ch2 sections 1,2,3: 2.1-6,16,38, 2.2-4,10, 2.3-9,27+Escape velocity (357.6 K, pdf, 29 Jan 2006)
    Text: ch2 DEplot maple example 1 for exercises 2.2, 2.3 (0.7 K, txt, 07 Sep 2009)
    Text: ch2 DEplot maple example 2 for exercises 2.2, 2.3 (0.7 K, txt, 07 Sep 2009)

html: Problem notes S2012 (5.2 K, html, 08 Apr 2012)
Examples and Applications
  Growth-Decay model y'=ky and its algebraic model y=y(0)exp(kx).
   Pharmokinetics of drug transport [ibuprofen]
   Pollution models.
     Three lake pollution model [Erie, Huron, Ontario].
   Brine tanks.
     One-tank model.
     Two-tank and three-tank models.
     Recycled brine tanks and limits of chapter 1 methods.
   Linear cascades and how to solve them.
     Method 1: Linear integrating factor method.
     Method 2: Superposition and equilibrium solutions for
       constant-coefficient y'+py=q. Uses the shortcut for
       homogeneous DE y'+py=0.
   Separation of variables
      The equation y'=7y(y-13), y(0)=17
      F(x) = 7, G(y) = y(y-13)
      Separated form y'/G(y) = F(x)
      Answer check using the Verhulst solution
          P(t) = aP_0/(bP_0 + (a-b P_0)exp(-at))
      Separation of variables details.
      Review of partial fractions.
      Partial fraction details for 1/((u(u-13)) = A/u + B/(u-13)
    References for 2.1, 2.2, 2.3. Includes the rabbit problem, partial fraction examples, phase diagram illustrations.
    Slides: Autonomous DE (102.3 K, pdf, 03 Mar 2012)
    Manuscript: Verhulst logistic equation (115.5 K, pdf, 02 Oct 2009)
    Manuscript: Phase Line and Bifurcation Diagrams. Includes Stability, Funnel, Spout, and bifurcation (227.4 K, pdf, 07 Sep 2009)
    Transparencies: ch2 sections 1,2,3: 2.1-6,16,38, 2.2-4,10, 2.3-9,27+Escape velocity (357.6 K, pdf, 29 Jan 2006)
    Text: ch2 DEplot maple example 1 for exercises 2.2, 2.3 (0.7 K, txt, 07 Sep 2009)
    Text: ch2 DEplot maple example 2 for exercises 2.2, 2.3 (0.7 K, txt, 07 Sep 2009)
    References for linear applications
    Manuscript: Applications of linear DE (374.2 K, pdf, 28 Jul 2009)
    Slides: Brink tanks (95.3 K, pdf, 04 Mar 2012)
    Slides: Home heating (109.8 K, pdf, 04 Mar 2012)
    Manuscript: Systems theory and examples (785.8 K, pdf, 16 Nov 2008)

Jan 25 and 27: Newton Kinematic Models. Projectiles. Problem. Section 2.3.

Drill and Review
  Phase diagram for y'=y^2(y^2-4)
     Phase line diagram
     Threaded curves
     Labels: stable, unstable, funnel, spout, node

  Phase line diagrams.
  Phase diagram.
Newton's force and friction models
  Isaac Newton ascent and descent kinematic models.
    Free fall with no air resistance F=0.
    Linear air resistance models F=kx'.
    Non-linear air resistance models F=k|x'|^2.

The tennis ball problem. Does it take longer to rise or longer to fall?
Slides: Newton kinematics with air resistance. Projectiles. (137.1 K, pdf, 03 Mar 2012)
Jules Verne problem. A rocket from the earth to the moon.
Slides: Jules Verne Problem (127.3 K, pdf, 03 Mar 2012)
Reading assignment: proofs of 2.3 theorems in the textbook and derivation of details for the rise and fall equations with air resistance.
Problem notes for 2.3-10. 2.3-20 (2.8 K, txt, 28 Jan 2012)
  Partial fractions.
   DEFINITION: partial fraction=constant/polynomial with exactly one root
   THEOREM: P(x)/q(x) = a sum of partial fractions
   Finding the coefficients.
     Method of sampling
       clear fractions, substitute samples, solve for A,B, ...
     Method of atoms
       clear fractions, multiply out and match powers, solve for A,B,...
     Heaviside's cover-up method
       partially clear fraction, substitute root, find one constant
  Separation of variable solutions with partial fractions.
  Exercise solutions to the problems due in 2.1.

Jan 26: Patrick

Exam 1 review, questions and examples on exam problems 1,2,3,4,5.
Sample Exam: Exam 1 key from F2010. See also S2010, exam 1.
Lecture on midterm 1 problems 4,5. Lecture on 2.2-10,18.
Maple Lab 2, problem 1 details [maple L2.1].