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.

Drill and ReviewPhase 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 modelsIsaac 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.

Problem notes for 2.3-10. 2.3-20 (1.8 K, txt, 24 Jan 2010)

Exam 1 review, questions and examples on exam problems 1,2,3,4,5. Exam 1 date is Sep 23, 7:25am in WEB 103. Also possible is Sep 20 or 22, 12:50pm in JWB 335. 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].

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);: connect-the-dots graphic (11.2 K, jpg, 12 Sep 2010)JPG Image

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:: Problems ER-1, ER-2 (109.2 K, pdf, 17 Aug 2010)Exam Review

Second lecture on numerical methods.Rect, Trap, Simp rules from calculusIntroduction 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.

Euler, Heun, RK4 algorithmsNumerical 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 ReferencesThe 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. The last is a quadrature probelm, which appears only in lecture notes from class. 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)=1 with solution y=x^2+x+1.

Example for your study:Problem: y'=x+1, y(0)=1 It has a dot table with x=0, 0.25, 0.5, 0.75, 1 and y= 1, 1.25, 1.5625, 1.9375, 2.375. The exact solution y = 0.5(1+(x+1)^2) has values y=1, 1.28125, 1.625, 2.03125, 2.5000. Determine how the dot table was constructed and identify which rule, either Rect, Trap, or Simp, was applied.

- 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)

- References for numerical methods:

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)

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)

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 reportConfused 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.