Week 3: 08 Sep, 09 Sep, 10 Sep, 11 Sep,

# 2250 Lecture Record Week 3 F2009

Last Modified: October 02, 2009, 17:32 MDT.    Today: August 16, 2018, 00:43 MDT.

## 08 Sep: Autonomous Differential Equations and Phase Diagrams. Section 2.2

First lecture after the Sep 7 holiday.
Collected today, 1.5: 8, 18
Intro to Maple lab 1: Theory of equations review, including quadratic equations, Factor and root theorem, division algorithm, recovery of the quadratic from its roots.
Lecture on 2.1, 2.2:
1. Theory of autonomous DE y'=f(y),
2. Stability of equilibrium solutions.
3. Stable and unstable classification of equilibrium solutions.
4. funnel, spout, node,
5. phase line diagram, phase diagram,
6. Partial fraction methods.
How to construct Phase line diagrams, calculus tools, DE tools, partial fraction methods.
How to make phase diagram graphic.
References for 2.1, 2.2:
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)

## 09 Sep: Phase diagram. Calculus tools. Section 2.2.

Second Lecture on 2.1, 2.2:
1. Stability of equilibrium solutions.
2. Stable and unstable classification of equilibrium solutions.
3. funnel, spout, node,
4. phase line diagram, phase diagram,
How to construct Phase line diagrams, calculus tools, DE tools, partial fraction methods.
How to make a phase diagram graphic.
Collected today, 1.5: 20, 34
Midterm 1 sample exam is the S2009 exam, found at the course web site.
HTML: 2250 midterm exam samples F2009 (14.5 K, html, 18 Dec 2009)

## 09-10 Sep: Fusi and Richins

Exam 1 review, questions and examples on exam problems 1,2,3,4,5.
Exam 1 date is Sep 30 1-5pm or Oct 1, 7am. All Web 103.
Sample Exam: Exam 1 key from S2009. See also F2008, exam 1.
Answer Key: Exam 1, S2009, 7:30am (395.7 K, pdf, 02 Mar 2009)
Answer Key: Exam 1, S2009, 10:45am (310.2 K, pdf, 02 Mar 2009)
Answer Key: Exam 1, f2008, 7:30am (407.3 K, pdf, 18 Feb 2009)
Lecture on midterm 1 problems 4,5. Lecture on 2.2-10,18.
Maple Lab 2, problem 1 details [maple L2.1].

## 11 Sep: Newton Kinematic Models. Projectiles. Jules Verne Problem. Section 2.3.

Lecture: Introduction to Newton's force and friction models.
Isaac Newton ascent and descent kinematic models.
1. Free fall with no air resistance F=0.
2. Linear air resistance models F=kx'.
3. 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 (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)

## Introduction to Numerical Solutions for Quadrature Equations y'=F(x)

Example in the 12:25 class: y'=x+1, y(0)=1 with 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.
1. Making the a table by approximation of the integral of F(x).
2. Rectangular rule.
3. Dot table steps for h=0.1. Answers: (x,y) = (0,1), (0.1,1.1), (0.2,1.21).
4. Maple support for making a connect-the-dots graphic.
Introduction: 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] allows interchange between different versions of maple.