# 2250-4 12:25pm Lecture Record Week 1 F2010

Last Modified: May 05, 2011, 11:48 MDT.    Today: July 15, 2018, 13:29 MDT.

## 23 and 25 Aug: Intro by Mon-Wed teaching assistant.

```Michal Kordy, LCB Loft
phone and email in the syllabus.
Details for maple lab 1.
Discuss submitted work presentation ideas.
Drill, examples, questions.
Discuss problems sections 1.2, 1.3, 1.4.
Discussion of Exam review plan for the semester.

24 Aug: Details about exams and dailies. Intro to DE, section
1.1.

Topics
Fundamental theorem of calculus. Method of quadrature [integration method in Edwards-Penney].
Slides: Fundamental Theorem of Calculus, Method of quadrature, Example. (87.3 K, pdf, 06 Jan 2010)Three Fundamental Examples introduced: growth-decay, Newton Cooling, Verhulst population.
Slide: Three Examples (11.8 K, pdf, 28 Aug 2006)Background from precalculus, logs and exponentials. Decay Equation Derivation.
Transparencies: Background Log+exponential. Problem 1.2-2 by J.Lahti. Decay law derivation. (213.3 K, pdf, 23 Jul 2009)Black and Lahti presentations of problem 1.2-1 and 1.2-2. Lahti presentations problems 1.2-5,8,10.
Transparencies: Three Examples. Solved problems 1.2-1,2 by Tyson Black, Jennifer Lahti, GBG (257.0 K, pdf, 26 Aug 2010)Transparencies: Solved problems 1.2-5,8,10 by Jennifer Lahti, GBG (139.1 K, pdf, 26 Aug 2010)Manuscript: Fundamentals, exponential modeling, applications, differential equations, direction fields, phase line, bifurcation, computing, existence (1122.3 K, pdf, 05 May 2011)
Textbook reference: Sections 1.1, 1.2.
Example for problem 1.2-1, similar to 1.2-2.
Panels 1 and 2 in the answer check for an initial value
problem like 1.2-2: y'=(x-2)^2, y(2)=1.
Answer checks. Proof that "0=1" and logic errors in presentations.
Maple tutorials start next week. Maple lab 1 is due soon, please
Maple: Lab 1,
Introduction (215.2 K, pdf, 17 Aug 2010)More on the method of quadrature:
Manuscript: The method of quadrature (with drill problems). (125.8 K, pdf, 26 Aug 2008)
Week 1 Syllabus, Format Suggestions, GradeSheet, Maple Lab 1
2250-4: Syllabus F2010 (150.8 K, pdf, 23 Aug 2010)2250: How to improve written work (41.5 K, pdf, 01 Jan 2010)2250-4: Gradesheet F2010.
Record-keeping system. (75.5 K, pdf, 24 Aug 2010)Maple: Lab 1, Introduction (215.2 K, pdf, 17 Aug 2010)

Week 1 references (documents, slides)
Slides: Fundamental Theorem of Calculus, Method of quadrature, Example. (87.3 K, pdf, 06 Jan 2010)Slide: Three Examples (11.8 K, pdf, 28 Aug 2006)Transparencies: Three Examples. Solved problems 1.2-1,2 by Tyson Black, Jennifer Lahti, GBG (257.0 K, pdf, 26 Aug 2010)Transparencies: Solved problems 1.2-5,8,10 by Jennifer Lahti, GBG (139.1 K, pdf, 26 Aug 2010)Transparencies: Background Log+exponential. Problem 1.2-2 by J.Lahti. Decay law derivation. (213.3 K, pdf, 23 Jul 2009)Manuscript: The method of quadrature (with drill problems). (125.8 K, pdf, 26 Aug 2008)Manuscript: Direction fields (540.9 K, pdf, 05 Jan 2010)Transparencies: Direction field examples, isocline method. (219.5 K, pdf, 23 Jul 2009)Transparency: Zoomed copy of Edwards-Penney exercise 1.3-8, to be used for homework (102.2 K, jpg, 29 Aug 2008)Slides: Summary of Peano, Picard, Direction Fields. (247.3 K, pdf, 26 Jul 2009)Manuscript: Picard-Lindelof and Peano Existence theory. (125.9 K, pdf, 18 Jan 2006)Slides: Peano and Picard Theory (22.5 K, pdf, 17 Jan 2007)Slide: Picard-Lindelof and Peano Existence Example, similar to 1.3-14. (40.5 K, pdf, 20 Jan 2006)Text: Background material functions and continuity (1.3-14). (5.0 K, txt, 05 Jan 2010)

24 Aug: Quadrature. Direction fields. Section 1.3.
Projection: Tyson Black 1.2-1, Jennifer Lahti 1.2-2
Exercises 1.2-4, 1.2-6, 1.2-10 discussion.
Integration details and how to document them using handwritten
calculations like u-substitution, parts, tabular.
Maple integration methods are possible [later in the course].
Integral table methods.
Integration theory examples.
Method of quadrature: Using Parts, tables, maple.
Discuss exercise 1.2-2 and exercise 1.2-10.
Maple lab 1
Inverse FOIL, complete the square, quadratic formula.
Theory of Equations.
Factor and root theorems.
Division algorithm.
Rational root theorem.
Descartes rule of signs.
Fundamental theorem of algebra.

27 Aug: Quadrature. Direction fields. Section 1.3
Integration tables
The first 20 entries in the front cover of our textbook are
required background.
Compute the integral of du/(1+u^2), 2u du/(1+u^2).
Integrals of rational functions have answers:
polynomial + log + arctan + constant.
Euler's directional field visualization.
Tools for using Euler's idea, which reduces an initial value
problem to infinitely many graphics.
The Idea: Display the behavior of all solutions, without solving
the differential equation.
The rules:
1. Solutions don't cross.
2. Threaded solutions pass other solutions with tangent line slope
nearly matching the nearby solutions.
Discuss problem 1.3-8.
For problem 1.3-8, xerox at 200 percent the textbook exercise page, then
cut and and paste the figure. Draw threaded curves on this figure
according to the rules in the direction field document above. To save
the xerox work, please print this prepared copy:

Transparency: Zoomed copy of Edwards-Penney exercise 1.3-8, to be used for homework (102.2 K, jpg, 29 Aug 2008) Direction field reference:
Manuscript: Direction fields (540.9 K, pdf, 05 Jan 2010) Topics on Direction fields
Threading edge-to-edge solutions is based upon two rules
[explained in the manuscript]:
1. Solution curves don't cross, and
2. Threaded solution curves nearly match tangents of nearby
direction field arrows.

The Picard-Lindelof theorem and the Peano theorem are
found in these web references. The theorems appear in section 1.3 of the textbook, without names.
Slides: Summary of Peano, Picard, Direction Fields. (247.3 K, pdf, 26 Jul 2009)Manuscript: Picard-Lindelof and Peano Existence theory. (125.9 K, pdf, 18 Jan 2006)Slides: Peano and Picard Theory (22.5 K, pdf, 17 Jan 2007)Transparencies: Picard-Lindelof and Peano Existence [1.3-14]. (40.5 K, pdf, 20 Jan 2006)Text: Background material functions and continuity (1.3-14). (5.0 K, txt, 05 Jan 2010)
How to thread curves on a direction field: Exercise 1.3-8.
Picard-Peano Example
y'=3(y-1)^(2/3), y(0)=1, similar to 1.3-14, from Peano-Picard slide above.
Exercise 1.3-14:
Justifications in exercise 1.3-14 are made from background
material in the calculus, taken from the link above "Background ... continuity".

Distribution of maple lab 1.
No printed maple lab 1? Print a copy from here:
Maple: Lab 1,
Introduction (215.2 K, pdf, 17 Aug 2010)
27 Aug: Theory and Examples for Separable Equations,
sections 1.4, 2.1

Review Topics
Drill: Direction fields.
Picard and Peano Theorems.
Question. We draw threaded solutions from some dot in the graphic.
How do we choose the dots? What do they represent?
Question. What does dy/dx=f(x,y), y(x0)=y0 have to do with threaded curves?
True and false trig formulas:
arctan(tan(theta))=theta  [false],
tan(arctan(x))=x [true].
Start topic of partial fractions, to be applied again in 2.1-2.2.
To be studied: Heaviside's method. Sampling method [a Fail-safe method].
Definition: A partial fraction is a constant divided by a polynomial
with exactly one root, that is, c/(x-r)^k. The root can be real or complex.

Definition of separable DE.
Examples: 1.4-6,12,18.
See the web site Problem Notes 1.4 for complete answers and
methods.

html: Problem notes F2010 (4.6 K, html, 26 Nov 2010)

Some separability tests.
Slides: Separable DE method. Tests I, II, III. Equilibrium solutions (110.2 K, pdf, 23 Jan 2010)
References for separable DE.
Slides: Separable DE method. Tests I, II, III. Equilibrium solutions (110.2 K, pdf, 23 Jan 2010)Manuscript: Method of quadrature (125.8 K, pdf, 26 Aug 2008)Manuscript: Separable Equations (171.3 K, pdf, 31 Aug 2009)Slides: Partial fraction theory (121.5 K, pdf, 30 Aug 2009)Manuscript: Heaviside's coverup method partial fractions (186.8 K, pdf, 20 Oct 2009)Text: How to do a maple answer check for y'=y+2x (0.2 K, txt, 27 Jan 2005)Transparencies: Section 1.4  and 1.5 Exercises (465.0 K, pdf, 26 Aug 2003)

Theory of separable equations section 1.4.
Separation test:
Define F(x)=f(x,y0)/f(x0,y0),
G(y)=f(x0,y),
then FG=f if and only if y'=f(x,y) is separable.
Non-Separable Test
f_x/f depends on y ==> y'=f(x,y) not separable
f_y/f depends on x ==> y'=f(x,y) not separable
Basic theory of y'=F(x)G(y):
y(x) = H^(-1)( C1 + int(F)),
H(u)=int(1/G,u0..u).
Solutions y=constant are called equilibrium solutions.
Find them using G(c)=0.
Non-equilibrium solutions arise from y'/G(y)=F(x) and a

Implicit and explicit solutions.
Discussion of answer checks for implicit solutions and also
explicit solutions.
Troubles with explicit solutions of y'= 3 sqrt(xy) [1.4-6].
Separable DE with no equilibrium solutions.
Separable DE with infinitely many equilibrium solutions.
The list of answers to a separable DE.
Influence of an initial condition to extract just one solution
formula from the list.
Examples for Midterm 1 problem 2:
y'=x+y, y'=x+y^2, y'=x^2+y^2
Example 1: Show that y'=x+y is not separable using TEST I or II
(partial derivative tests).
Example 2: Find the factorization f=F(x)G(y) for y'=f(x,y),
given
(1) f(x,y)=2xy+4y+3x+6 [ans: F=x+2, G=2y+3].
(2) f(x,y)=(1-x^2+y^2-x^2y^2)/x^2 [ans: F=(1-x^2)/x^2, G=1+y^2].

References for separable DE.
Manuscript: Method of quadrature (125.8 K, pdf, 26 Aug 2008)Slides: Separable DE method. Tests I, II, III. Equilibrium solutions (110.2 K, pdf, 23 Jan 2010)Manuscript: Separable Equations (171.3 K, pdf, 31 Aug 2009)Text: How to do a maple answer check for y'=y+2x (0.2 K, txt, 27 Jan 2005)Transparencies: Section 1.4 and 1.5 Exercises (465.0 K, pdf, 26 Aug 2003)
Reading on partial fractions. We study (1) sampling, (2) method of atoms, (3) Heaviside cover-up.
Slides: Partial Fraction Theory (121.5 K, pdf, 30 Aug 2009)Manuscript: Heaviside coverup partial fraction method (152.1 K, pdf, 07 Aug 2009)Manuscript: Heaviside's method and Laplace theory (186.8 K, pdf, 20 Oct 2009)

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