Review: Problem 1.5-34The expected model is x'=1/4-x/16, x(0)=20, using units of millions of cubic feet. The answer is x(t)=4+16 exp(-t/16). Model Derivation Law: x'=input rate - output rate. Definition: concentration == amt/volume. Use of percentages 0.25% concentration means 0.25/100 concentration

Examples and ApplicationsGrowth-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.

Intro to Maple lab 1Theory of equations review quadratic equations, Factor and root theorem, division algorithm, recovery of the quadratic from its roots.

Lecture on 2.1, 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 Definition: phase line diagram, phase diagram, Calculus tools: f'(x) pos/neg ==> increasing/decreasing DE tools: solutions don't cross, tangent matching 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 Partial fraction methods and solution formulas

- 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 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 four problems due in 2.1, 2.2. 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 date is Feb 17, 2-4pm in WEB 104 or Feb 18, 6:50am in JTB 140.

Sample Exam: Exam 1 key from F2009. See also S2009, 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'=x+1, y(0)=1 Symbolic solution y=0.5 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), (0.1,1.1), (0.2,1.21). Maple support for making a connect-the-dots graphic. Next week: Maple sessions with Rect, Trap, Simp rules: online code.

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 Spring Break.: Problems ER-1, ER-2 (46.9 K, pdf, 01 Jan 2010)Exam Review

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)=3 with solution y=x^2+x+3. 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.

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.

- Introduction: Maple Labs 3 and 4, due after Spring Break.

Maple lab 3 S2010. Numerical DE (82.5 K, pdf, 01 Jan 2010)

Maple lab 4 S2010. Numerical DE (78.9 K, pdf, 01 Jan 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)

S2010 notes on numerical DE report for Ch2 Ex 10 (49.5 K, pdf, 01 Jan 2010)

S2010 notes on numerical DE report for Ch2 Ex 12 (64.4 K, pdf, 01 Jan 2010)

S2010 notes on numerical DE report for Ch2 Ex 4 (49.5 K, pdf, 01 Jan 2010)

S2010 notes on numerical DE report for Ch2 Ex 6 (65.9 K, pdf, 01 Jan 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)

Confused about what to put in your L3.1 report? Do the same as what appears in the sample report for 2.4-3 (link below). 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.

Sample Report for 2.4-3 (0.0 K, pdf, 31 Dec 1969)

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.

Euler, Heun, RK4 algorithmsComputer implementation in maple Geometric and algebraic ideas in the derivations. Numerical Integration Numerical Solutions of DE RECT Euler TRAP Heun [modified Euler] SIMP Runge-Kutta 4 [RK4] Numerical work maple L3.1, L3.2, L3.3, L4.1, L4.2, L4.3 will be submitted after Spring Break. No numerical problems from ch 2 are assigned. All discussion of maple programs will be based in the TA session [Cox and Murphy]. 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.

ExamplesWeb references contain two kinds of examples. The first three are quadrature problems dy/dx=F(x). The fourth 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'=2x+1, y(0)=3 with solution y=x^2+x+3. y'=1-x-y, y(0)=3, solution y=2-x+exp(-x).