2280 12:55pm Lectures S2015, Week 1

Last Modified: January 15, 2015, 12:34 MST.    Today: December 14, 2018, 01:27 MST.

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  Sections 1.2, 1.3, 1.4, 1.5
  The textbook topics, definitions, examples and theorems
Edwards-Penney 1.2, 1.3, 1.4, 1.5 (14.6 K, txt, 12 Jan 2015)
PDF: Week 1 Examples (131.9 K, pdf, 11 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 (106.7 K, pdf, 27 Dec 2014)
Take-Home Exam: Quiz1 due next week (114.9 K, pdf, 20 Jan 2015)
Homework: Package HW1 due next week (44.9 K, pdf, 26 Jan 2015)
Problem Notes: Chapter 1 (15.9 K, txt, 09 Dec 2014)

Week 1, Sections 1.1,1.2,1.3,1.4.

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

 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 theory
Manuscript: 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. Answer Check. 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, Bookmark
2280 8:05: Syllabus S2015 (257.1 K, pdf, 10 Jan 2015)
2280: How to improve written work (79.0 K, pdf, 08 Dec 2014)
2280 8:05: Book Mark S2015. (81.5 K, pdf, 13 Jan 2015)

Mon-Wed: Quadrature. Section 1.2.

Projection: Handwritten exercise solutions: Tyson Black 1.2-1, Jennifer Lahti 1.2-2
Transparencies: Solved problems 1.2-1 by Tyson Black, 1.2-2 by Jennifer Lahti (257.0 K, pdf, 25 Aug 2010) Topics on Quadrature 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). (237.7 K, pdf, 06 Jan 2014) Quadratic equations. 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. Drill: Quadrature 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. Two Threading Rules. 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)

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 fractions
Slides: 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 1.4 for complete answers and methods. Some separability tests. Read the first slide link below, Tests I, II, III.
html: Problem notes S2015 (2.5 K, html, 23 Dec 2014) 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 (237.7 K, pdf, 06 Jan 2014)
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 quadrature step. 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. Answer Checks and Key Examples. Discussion of answer checks implicit solution ln|y|=2x+c for y'=2y explicit solution y = C exp(2x) for y'=2y Answer check for y'= 3 sqrt(xy) [1.4-6]. Answer checks for midterm examples y'=x+y, y'=x+y^2, y'=x^2+y^2