# 2250-1 7:30am Lecture Record Week 1 S2010

Last Modified: January 15, 2010, 06:11 MST.    Today: March 23, 2018, 05:09 MDT.

## 11 Jan: Details about exams and dailies. Intro to DE, section 1.1.

Topics
1. 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)
2. Three Fundamental Examples introduced: growth-decay, Newton Cooling, Verhulst population.
Slide: Three Examples (11.8 K, pdf, 28 Aug 2006)
3. 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)
4. Black and Lahti presentations of problem 1.2-1 and 1.2-2.
Transparencies: Three Examples. Solved problems 1.2-1,2,5 by Tyson Black, Jennifer Lahti, GBG (292.0 K, pdf, 23 Jul 2009)

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 print it from the link
Maple: Lab 1, Introduction (112.5 K, pdf, 01 Jan 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
2250-1: Syllabus S2010 (150.5 K, pdf, 11 Jan 2010)
2250: How to improve written work (41.5 K, pdf, 01 Jan 2010)
2250-1: Gradesheet S2010. Record-keeping system. (80.3 K, pdf, 03 Jan 2010)
Maple: Lab 1, Introduction (112.5 K, pdf, 01 Jan 2010)

## 12 Jan: Quadrature. Direction fields. Section 1.3.

Start topic of partial fractions, to be applied again in 2.1-2.2. Heaviside's method. Sampling method [a Fail-safe method].
```Topics on Quadrature
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.
Integral table methods.
Integration theory examples.
Method of quadrature: Using Parts, tables, maple.
Discuss exercise 1.2-8.
```
``` 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.
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. Use 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.
```

## 13 Jan: Existence-Uniqueness: Picard and Peano theorems. Section 1.4.

```Collected in class Page 16, 1.2:  4, 6.
Drill: How to thread curves on a direction field.
Exercise 1.3-8.
Drill: Picard-Lindelof Theorem, Peano Theorem.
Picard-Peano Example
y'=3(y-1)^(2/3), y(0)=1, similar to 1.3-14.
Exercise 1.3-14:
Justifications in exercise 1.3-14 are made from background
material in the calculus:
```

Text: Background material functions and continuity (1.3-14). (5.0 K, txt, 05 Jan 2010)
Distribution of maple lab 1. No printed maple lab 1? Print a copy from here:
Maple: Lab 1, Introduction (112.5 K, pdf, 01 Jan 2010)

## 14 Jan: Intro by Ben Murphy and Charles Cox.

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.

## 15 Jan: Theory and Examples for Separable Equations, sections 1.4, 2.1

```Review Topics
Drill: Direction fields.
Picard and Peano Theorems.
We draw threaded solutions from some dot in the graphic. How
do we choose the dots? What do they represent?
What does dy/dx=f(x,y), y(x0)=y0 have to do with threaded
curves?
integral of du/(1+u^2), 2u du/(1+u^2).
True and false trig formulas:
arctan(tan(theta))=theta  [false],
tan(arctan(x))=x [true].
```
```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 S2010 (4.4 K, html, 31 Jan 2010)
Some separability tests.
Slides: Separable DE method. Tests I, II, III. Equilibrium solutions (110.2 K, pdf, 23 Jan 2010)
``` 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].
```
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)