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2250-4 12:25pm Lecture Record Week 3 F2010

Last Modified: September 12, 2010, 08:16 MDT.    Today: December 10, 2017, 21:16 MST.

Week 3, Sep 6 to 10: Sections 2.2, 2.3, 2.4

Sep 8 and 9: Laura and Michal

Exam 1 review, questions and examples on exam problems 1,2,3,4,5.
Exam 1 date is Sep 20 or 22, 12:50pm in JWB 335.  Also possible is Sep 23, 7:25am in WEB 103.
Sample Exam: Exam 1 key from S2010. See also F2009, exam 1.
Lecture on midterm 1 problems 4,5. Lecture on 2.2-10,18.
Maple Lab 2, problem 1 details [maple L2.1].

Sep 7: Autonomous Differential Equations and Phase Diagrams. Section 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 (69.9 K, pdf, 03 Sep 2009)
    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)

Sep 7: Newton Kinematic Models. Projectiles. Problem. Section 2.3.

Drill and Review
  Phase diagram for y'=y(1-y)(2-y)
     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. (109.3 K, pdf, 29 Aug 2009)
Jules Verne problem. A rocket from the earth to the moon.
Slides: Jules Verne Problem (91.9 K, pdf, 27 Jan 2010)
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 (1.8 K, txt, 24 Jan 2010)

Sep 9: Numerical Solutions for y'=F(x)

Numerical Solution of y'=F(x)
  Example: y'=2x+1, y(0)=1
    Symbolic solution y=x^2 + x + 1.
    Dot table. Connect the dots graphic.
    How to draw a graphic without knowing the solution equation for y.
    Making the dot table by approximation of the integral of F(x).
      Rectangular rule.
      Dot table steps for h=0.1.
        Answers: (x,y) = [0, 1], [.1, 1.1], [.2, 1.22], [.3, 1.36], [.4, 1.52],
        [.5, 1.70], [.6, 1.90], [.7, 2.12], [.8, 2.36], [.9, 2.62], [1.0, 2.90]
      The exact answers for y(x)=x^2+x+1 are
        (x,y) = [0., 1.], [.1, 1.11], [.2, 1.24], [.3, 1.39], [.4, 1.56],
        [.5, 1.75], [.6, 1.96], [.7, 2.19], [.8, 2.44], [.9, 2.71], [1.0, 3.00]
  
 Maple support for making a connect-the-dots graphic.
        Example: L:=[[0., 1.], [2,3], [3,-1], [4,4]]; plot(L);
JPG Image: connect-the-dots graphic (11.2 K, jpg, 12 Sep 2010)
 Maple code for the RECT rule, applied to quadrature problem y'=2x+1, y(0)=1.
    # Quadrature Problem y'=F(x), y(x0)=y0.
    # Group 1, initialize.
    F:=x->2*x+1:
    x0:=0:y0:=1:h:=0.1:Dots:=[x0,y0]:n:=10:

   # Group 2, repeat n times. RECT rule.
     for i from 1 to n do
       Y:=y0+h*F(x0);
       x0:=x0+h:y0:=Y:Dots:=Dots,[x0,y0];
     od:

   # Group 3, display dots and plot.
     Dots;
     plot([Dots]);
Numerical Solution of y'=f(x,y)
   Two problems will be studied, in maple labs 3, 4.
   First problem
      y' = -2xy, y(0)=2
      Symbolic solution y = 2 exp(-x^2)
   Second problem
      y' = (1/2)(y-1)^2, y(0)=2
      Symbolic solution y = (x-4)/(x-2)
   The work begins in exam review problems ER-1, ER-2, both due
   before the first midterm exam. The maple numerical work is due
   much later, after Semester Break. Here's the statements for the
   exam review problems, which review chapter 1 methods to find the
   symbolic solutions:
Exam Review: Problems ER-1, ER-2 (109.2 K, pdf, 17 Aug 2010)
Rect, Trap, Simp rules from calculus
   Introduction to the Euler, Heun, RK4 rules from this course.
   Example: y'=3x^2-1, y(0)=2 with solution y=x^3-x+2.
   Example: y'=2x+1, y(0)=1 with solution y=x^2+x+1.
   Dot tables,  connect the dots graphic.
   How to draw a graphic without knowing the solution equation for y.
     Key example y'=sqrt(x)exp(x^2), y(0)=2.
     Challenge: Can you integrate sqrt(x) exp(x^2)?
   Making the dot table by approximation of the integral of F(x).
   Rect, Trap, Simp rules and their accuracy of 1,2,4 digits resp.
    Make our own copy from the web: Maple Labs 3 and 4, due after Semester Break.
    Maple lab 3 F2010. Numerical DE (152.5 K, pdf, 17 Aug 2010)
    Maple lab 4 F2010. Numerical DE (138.2 K, pdf, 17 Aug 2010)

Sep 9: Numerical Solutions for y'=f(x,y)

Second lecture on numerical methods: Preview of next week's lecture.
Euler, Heun, RK4 algorithms
   Computer implementation.
   Geometric and algebraic ideas in the derivations.
 Numerical work maple L3.1, L3.2, L3.3, L4.1, L4.2, L4.3 will be
 submitted after Semester Break. All discussion of maple programs will be
 based in the TA session [Laura and Michal]. There will be one additional
 presentation of maple lab details in the main lecture. The examples used
 in maple labs 3, 4 are the same as those in exam review problems ER-1,
 ER-2. Each has form dy/dx=f(x,y) and requires a non-quadrature
 algorithm, e.g., Euler, Heun, RK4. The examples in maple L3, L4:
    y'=-2xy, y(0)=2, solution y=2exp(-x^2)
    y'=(1/2)(y-1)^2, y(0)=2, solution y=(x-4)/(x-2).
 Web references contain three examples. The first two are quadrature
 problems dy/dx=F(x). The third is of the form dy/dx=f(x,y), which
 requires a non-quadrature algorithm like Euler, Heun, RK4.
    y'=3x^2-1, y(0)=2, solution y=x^3-x+2
    y'=exp(x^2), y(0)=2, solution y=2+int(exp(t^2),t=0..x).
    y'=1-x-y, y(0)=3, solution y=2-x+exp(-x).
    y'=2x+1, y(0)=3 with solution y=x^2+x+3. [lecture notes only]
    References for numerical methods:
    Manuscript: Numerical methods manuscript (112.1 K, pdf, 03 Sep 2009)
    Text: Maple L3 snips F2010 (maple text) (5.0 K, txt, 24 May 2007)
    Maple Worksheet: Maple L3 snips F2010 (maple .mws) (6.6 K, mws, 25 May 2007)
    Text: Maple code for maple labs 3 and 4 (5.1 K, txt, 17 Aug 2010)
    Maple Worksheet: Sample maple code for Euler, Heun, RK4 (1.9 K, mws, 21 Aug 2010)
    Maple Worksheet: Sample maple code for exact/error reporting (2.1 K, mws, 21 Aug 2010)
    How to use maple at home (4.0 K, txt, 06 Jan 2010)
    Maple lab 3 symbolic solution, ER-1 solution. (184.6 K, jpg, 08 Feb 2008)
    Transparencies: Sample Report for 2.4-3. Includes symbolic solution report. (175.9 K, pdf, 02 Jan 2010)
    F2010 notes on numerical DE report for Ch2 Ex 10 (97.9 K, pdf, 17 Aug 2010)
    F2010 notes on numerical DE report for Ch2 Ex 12 (108.4 K, pdf, 17 Aug 2010)
    F2010 notes on numerical DE report for Ch2 Ex 4 (97.8 K, pdf, 17 Aug 2010)
    F2010 notes on numerical DE report for Ch2 Ex 6 (125.7 K, pdf, 17 Aug 2010)
    Sample Report for 2.4-3 (175.9 K, pdf, 02 Jan 2010)
    Transparencies: ch2 Numerical Exercises 2.4-5,2.5-5,2.6-5 plus Rect, Trap, Simp rules (219.5 K, pdf, 29 Jan 2006)
The work for book sections 2.4, 2.5, 2.6 is in maple lab 3 and maple lab 4.
The numerical work using Euler, Heun, RK4 appears in L3.1, L3.2, L3.3.
       The actual symbolic solution derivation and answer check were
       submitted as Exam Review ER-1. Confused? Follow the details in
       the next link, which duplicates what was done in ER-1.
Sample symbolic solution report for 2.4-3 (22.6 K, pdf, 19 Sep 2006)
Maple lab 3 report
        Confused about what to put in your L3.1 report? Do the same
        as what appears in the sample report for 2.4-3. 
Sample Report for 2.4-3 (175.9 K, pdf, 02 Jan 2010)
        Include  the hand answer check. Include the maple code appendix. Then fill
        in the table in maple Lab 3, by hand.
        The example shows a  hand answer check and the maple code appendix.
Download all .mws maple worksheets to disk, then run the worksheet in xmaple.
       In Mozilla firefox, save to disk using right-mouseclick and then "Save link as...".
       Some browsers require SHIFT and then mouse-click. Open the saved file in
       xmaple or maple.

       Extension .mws [or .mpl] allows interchange between
       different versions of maple. Mouse copies of the worksheet pasted into email
       allow easy transfer of code between versions of maple.