Week 4: 14 Sep, 15 Sep, 16 Sep, 17 Sep, 18 Sep,

# 2250 Lecture Record Week 4 F2009

Last Modified: October 02, 2009, 17:34 MDT.    Today: March 21, 2018, 20:11 MDT.

## 14 Sep: Partial fractions. Phase diagrams. Problem session Chapter 2.

```  The Jules Verne problem [slides]. Finished last week in 12:25, finished today in 7:30.
Partial fractions.
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, report 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.
Postponed:
Verhulst models with harvesting term.
Differences in phase diagrams of non-autonomous systems.
What new things we see in computer graphics of such models.
```
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 (99.7 K, pdf, 27 Jan 2008)
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)

## 15 Sep: Euler, Heun and RK4 Algorithms for dy/dx = f(x,y), y(x0)=y0.

Lecture on symbolic solution exam review problems ER-1, ER-2 [identical to maple L3.1, L4.1], which are due next week. Done in 12:25 last week.
Exam Review: Problem statements ER-1, ER-2 (47.0 K, pdf, 31 Jan 2009)
Transparency: Sample solution ER-1 [same as L3.1] (184.6 K, jpg, 08 Feb 2008)
Introduction to 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)=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.
1. Key example y'=sqrt(x)exp(x^2), y(0)=2.
Challenge: Can you integrate sqrt(x) exp(x^2)?
2. Making the dot table by approximation of the integral of F(x).
3. Rect, Trap, Simp rules and their accuracy of 1,2,4 digits resp.
1. The problem y'=x+1, y(0)=1 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.
2. The exact solution y=1/2+(x+1)^2/2 has values y=1, 1.28125, 1.625, 2.03125, 2.5000.
3. Determine how the dot table was constructed and identify which rule [Rect, Trap, Simp] was applied.

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

Second lecture on numerical methods.
Lecture on Euler, Heun, RK4 algorithms. Computer implementation. Geometric and algebraic ideas in the derivations.
Numerical work maple L3.2, L3.3, L3.4, L4.2, L4.3, L4.4 will be submitted after Fall Break.
1. All discussion of maple programs will be based in the TA session [Fusi and Richins].
2. 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 ER-1, ER-2. Each has form dy/dx=f(x,y) and requires a non-quadrature algorithm, e.g., Euler, Heun, RK4.
1. y'=-2xy, y(0)=2, solution y=2exp(-x^2)
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.
1. y'=3x^2-1, y(0)=2, solution y=x^3-x+2
2. y'=exp(x^2), y(0)=2, solution y=2+int(exp(t^2),t=0..x).
3. y'=1-x-y, y(0)=3, solution y=2-x+exp(-x).
4. y'=2x+1, y(0)=3 with solution y=x^2+x+3. [lecture notes only]
Maple Labs 3 and 4, due after Fall Break.
Maple lab 3 F2009. Numerical DE (81.9 K, pdf, 19 Jul 2009)
Maple lab 4 F2009. Numerical DE (78.2 K, pdf, 19 Jul 2009)

The work for 2.4, 2.5, 2.6 is in maple lab 3 and maple lab 4.
Submitting Exam Review ER-1 earns credit for L3.1. Do not submit both ER-1 and L3.1!
The numerical work using Euler, Heun, RK4 appears in L3.2, L3.3, L3.4.
Confused about what to put in your L3.2 report? Do the same as what appears in the sample report for 2.4-3.
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 allow interchange between different versions of maple.

## 17 Sep: Review and Chapters 1-2. Drill

Fusi and Richins review Sample Midterm problems 1-5.
More details on maple lab 2.
How to present the solutions to the exam review problems ER-1 and ER-2. These are identical to math-only [no computer] problems L3.1, L4.1.

## 18 Sep: Linear Algebraic Equations. No matrices. Section 3.1.

Due today: 2.2-10, 2.2-14.
Lecture: 3.1, 3.2, frame sequences, combo, swap, multiply, plane and space geometry,
three possibilities, method of elimination, parameters,
Differential equations example and unique solution of a 2x2 system. See problem notes section 3.1:
html: Problem notes F2009 (4.0 K, html, 22 Sep 2009)
Prepare 3.1 problems for next collection. Lecture on Maple lab 2 problem 1.