#### 2250 8:05am Lectures S2015, Week 1

Last Modified: January 19, 2015, 20:57 MST.    Today: September 24, 2018, 10:55 MDT.

```Topics
Sections 1.2, 1.3, 1.4, 1.5
The textbook topics, definitions, examples and theoremsEdwards-Penney 1.2, 1.3, 1.4, 1.5 (14.6 K, txt, 12 Jan 2015)Examples: Week 1 (337.6 K, png, 25 Jan 2015)
WEEK 1
======
1.1; differential equation, mathematical model.
1.2; integral as a general or particular solution.
1.3; slope field.
1.4; separable differential equation.
Take-Home Exam: Sample Quiz1 (113.8 K, pdf, 12 Jan 2014)Take-Home Exam: Quiz1 due next week (116.0 K, pdf, 20 Jan 2015)Homework: Package HW1 due next week (36.6 K, pdf, 29 Nov 2014)Problem Notes: Chapter 1 (2.8 K, html, 01 Jan 2015)```

### Monday: Details about exams and homework. Intro to DE, sections 1.1 and 1.2.

```Topics
Fundamental theorem of calculus.
Method of quadrature [integration method in Edwards-Penney].Slides: Fundamental Theorem of Calculus, Method of quadrature, Example, 2-Panel answer check. (131.0 K, pdf, 09 Jan 2015)
Exponential modeling, first order applications, Peano and Picard theoryManuscript: Fundamentals, exponential modeling, applications, differential equations, direction fields, phase line, bifurcation, computing, existence (1432.3 K, pdf, 03 Dec 2014)
Three Fundamental Examples introduced: growth-decay, Newton Cooling,
Verhulst population.Slide: Three Examples (11.8 K, pdf, 09 Dec 2012)
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, 22 Jul 2009)
Black and Lahti presentations of problem 1.2-1 and 1.2-2.Transparencies: Three Examples. Solved problems 1.2-1 by Tyson Black, 1.2-2 by Jennifer Lahti (257.0 K, pdf, 25 Aug 2010)
Lahti presentations of problems 1.2-5, 1.2-8, 1.2-10.Transparencies: Solved problems 1.2-5,8,10 by Jennifer Lahti (139.1 K, pdf, 25 Aug 2010)
Example: Problem 1.2-2.
Solve y'=(x-2)^2, y(2)=1.
Panels 1 and 2 for the initial value
y'=(x-2)^2,y(2)=1.
Proof that "0=1".
Non-reversible steps and logic errors in presentations.Slides: Fundamental Theorem of Calculus, Method of quadrature, Example, 2-Panel answer check. (131.0 K, pdf, 09 Jan 2015)
Syllabus, Writing Suggestions, Gradesheet2250 8:05: Syllabus S2015 (262.3 K, pdf, 18 Mar 2015)2250: How to improve written work (79.1 K, pdf, 02 Dec 2014)2250 8:05: Book Mark S2015. (76.8 K, pdf, 18 Mar 2015)```

```Projection: Handwritten exercise solutions: Tyson Black 1.2-1, Jennifer Lahti 1.2-2Transparencies: Solved problems 1.2-1 by Tyson Black, 1.2-2 by Jennifer Lahti (257.0 K, pdf, 25 Aug 2010)
Exercises 1.2-4, 1.2-6, 1.2-10 discussion.
Integration details and how to document them
u-substitution, parts, tabular.
Maple integration methods are possible [later in the course].
Integral table methods.
Integration theory examples.Slides: Fundamental Theorem of Calculus, Method of quadrature, Example. (131.0 K, pdf, 09 Jan 2015)
Method of quadrature: Using Parts, tables, maple.
Discuss exercise 1.2-2 and exercise 1.2-10.
Reference for the method of quadrature:Manuscript: The method of quadrature (with drill problems). (242.8 K, pdf, 09 Jan 2015)
Inverse FOIL, complete the square, quadratic formula.Slides: Theory of equations, quadratics. (78.1 K, pdf, 07 Dec 2014)
Theory of Equations.
Factor and root theorems.
Division algorithm.
Rational root theorem.
Descartes' rule of signs.
Fundamental theorem of algebra, order n has exactly n roots.
Integration techniques
u-substitution  (x+2)^3dx, x sin(x^2)dx, xdx/(x+1)
parts  xe^xdx, ln(x)dx
partial fractions  xdx/(x^4-1)
trig  sin(x)dx, sin(x)cos(x)dx, cos^2(x)dx
hyperbolic  sinh(x)dx
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.
```

### Wednesday: Direction fields. Peano and Picard. Section 1.3

```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 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, 09 Dec 2012)
Direction field references:Manuscript: Direction fields (656.2 K, pdf, 06 Jan 2014)Slides: Summary of Peano, Picard, Direction Fields. (293.7 K, pdf, 03 Mar 2012)PNG: Picard iterates example. (145.6 K, png, 15 Jan 2015)

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.
Picard and Peano Theorems
The Picard-Lindelof theorem and the Peano theorem are
found in the web references below. The theorems appear in
section 1.3 of the textbook, without names.
PEANO THEOREM [brief statement]
y'=f(x,y), y(x0)=y0 has at least one solution
provided f is continuous.
PICARD-LINDELOF THEOREM [brief statement]
y'=f(x,y), y(x0)=y0 has one and only one solution
provided f is continuously differentiable.
SOLUTION GEOMETRY
The Peano and Picard theorems conclude existence of a curve
y=y(x) AND ALSO a Box B with center (x0,y0). Curve y(x) crosses
the box edge-to-edge, from left to right (it does not exit the
top or bottom), passing through the center point (x0,y0).
Manuscript: Picard-Lindelof and Peano Existence theory. (304.2 K, pdf, 06 Jan 2014)Transparency: Picard-Lindelof and Peano Existence [1.3-14, Dirichlet]. (40.5 K, pdf, 20 Jan 2006)Text: Background material functions and continuity (1.3-14). (4.1 K, txt, 05 Dec 2012)
Remarks on Exercises 1.3
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".
Summary of 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].
Switches and Finite Blowup of Solutions

Differential equations y'=f(x,y) in which f is defined piecewise
with switches may have a unique solution. Differential equations with
f smooth have a unique solution, but the solution may blow up in finite time.
Here's two examples:PNG: Switch example, Blowup example (309.9 K, png, 14 Jan 2015)```

### Thursday: Intro by Thu teaching assistant.

```  Ziwen Zhu
LCB Loft
CANVAS: Zhu monitors messages sent via canvas
Discuss submitted work presentation ideas.
Attendance.
Introduce Lab 1 topics, distribute paper copy.
```

### Wed-Fri: Theory and Examples for Separable Equations, sections 1.4, 2.1

```Partial Fractions
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]. The method of atoms.
References on partial fractionsSlides: Partial Fraction Theory (148.6 K, pdf, 14 Dec 2014)Manuscript: Heaviside coverup partial fraction method (290.2 K, pdf, 07 Jan 2014)Manuscript: Heaviside's method and Laplace theory (352.3 K, pdf, 07 Jan 2014)
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 for complete answers and
methods.html: Problem notes S2015 (2.8 K, html, 01 Jan 2015)
Some separability tests.
References for separable DE.Slides: Separable DE method. Tests I, II, III. Equilibrium solutions (294.2 K, pdf, 13 Jan 2015)Manuscript: Method of quadrature (242.8 K, pdf, 09 Jan 2015)Manuscript: Separable Equations (314.8 K, pdf, 06 Jan 2014)Text: How to do a maple answer check for y'=y+2x (0.3 K, txt, 07 Jan 2014)Transparencies: Section 1.4 and 1.5 Exercises (465.0 K, pdf, 26 Aug 2003)
Theory of separable equations section 1.4.
Separable equations depend on partial fraction theory, reading below.
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
TEST I. f_x/f depends on y ==> y'=f(x,y) not separable
TEST II. f_y/f depends on x ==> y'=f(x,y) not separable
Review: 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.
Exercise 1.4-6: Trouble with explicit solutions of y'= 3 sqrt(xy)
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 of solutions.
Key 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
TEST I. f_x/f depends on y ==> y'=f(x,y) not separable
TEST II. f_y/f depends on x ==> y'=f(x,y) not separable
Example 2: Find the factorization f=F(x)G(y) for y'=f(x,y), given
(1) f(x,y)=2xy+4y+3x+6
(2) f(x,y)=(1-x^2+y^2-x^2y^2)/x^2
Answers: (1) F=x+2, G=2y+3; (2) F=(1-x^2)/x^2, G=1+y^2
Idea: F(x)=f(x,y)/G(y) implies F(x)=f(x,0)/G(0) for y=0, which
implies \$F(x)=f(x,0)\$ divided by some constant c. Backsub
implies G(y)=f(x,y)/F(x)= c f(x,y)/f(x,0). Cleverly select
y=y0 instead of y=0 for a general method.
Monday: Answer Checks and Key Examples.