Week 2: 31 Aug, 01 Sep, 02 Sep, 03 Sep, 04 Sep,

# 2250 Lecture Record Week 2 F2009

Last Modified: October 02, 2009, 04:33 MDT.    Today: September 24, 2018, 01:23 MDT.

## 31 Aug: Theory and Examples for Separable Equations, sections 1.4, 2.1

Collected in class, Page 26, 1.3-8.
Drill: Direction fields, Two Threading Rules. Picard and Peano Theorems.
Drill: We draw threaded solutions from some dot in the graphic. How do we choose the dots? What do they represent?
Drill: What does dy/dx=f(x,y), y(x0)=y0 have to do with threaded curves?
Drill: Quadrature, 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].
Solutions for 1.4-6,12,18. See also Problem Notes 1.4 at the web site.
Exercises Page 41, 1.4: 6, 12 are due next.
Theory of separable equations continued, section 1.4.
Separation test:
1. Define F(x)=f(x,y0)/f(x0,y0), G(y)=f(x0,y),
2. then FG=f if and only if y'=f(x,y) is separable.
3. Basic theory: y(x) = H^(-1)( C1 + int(F)), H(u)=int(1/G,u0..u).
4. 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.
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)

## 01 Sep: Theory of Linear First Order Differential Equations. Section 1.5.

Collected in class, Page 26, 1.3-14.
Drill on the variables separable method. Discuss remaining 1.4 exercises.
Lecture on Section 1.5, theory of linear DE y'=-P(x)y+Q(x).
Integrating factor, the fraction that replaces two-termed expression y'+py.
Classification of y'=f(x,y): quadrature, separable, linear [QSL].
1. Venn diagram of classes Q, S, L.
2. Examples of various types.
3. Test for quadrature (f_y=0)
4. Test for linear (f_y indep of y)

Linear integrating factor method 1.5. Application to y'+2y=1 and y'+y=e^x.
Examples: Testing linear DE y'=f(x,y) by f_y independent of y.
Examples: linear equations and non-linear equations.
Picard's theorem implies a linear DE has a unique solution.
Main theorem on linear DE and explicit general solution.

## 02 Sep: Linear Applications. Section 1.5

Collect in class Page 41, 1.4: 6, 12.
1. More about problem 1.5-34
Text: Problem notes on 1.5-34 (1.9 K, txt, 19 Jul 2009)
2. The expected model is x'=1/4-x/16, x(0)=20, using units of millions of cubic feet.
3. The answer is x(t)=4+16 exp(-t/16).
4. Model Derivation uses x'=input rate - output rate.
Definition of concentration == amt/volume.
Use of percentages in concentrations [0.25% concentration means 0.25/100 == concentration].
Superposition for y'+p(x)y=0. Superposition for y'+p(x)y=q(x).
Examples and Applications
1. Growth-Decay model y'=ky and its algebraic model y=y(0)exp(kx).
Pharmokinetics for drug transport [ibuprofen], brine tanks, pollution models.
2. One-tank model. Two-tank and three-tank models.
3. Recycled brine tanks and non-solvability by chapter 1 methods.
4. Three lake pollution model [Erie, Huron, Ontario].
5. Linear cascades and how to solve them.
1. Method 1: Linear integrating factor method.
2. Method 2: Superposition and equilibrium solutions for constant-coefficient y'+py=q. Uses a shortcut for growth-Decay DE y'+py=0

## 02-03 Sep: Fusi and Richins

Present problems 2, 3 of the midterm 1 sample [S2009 midterm 1 key].
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)
Questions on textbook sections 1.3, 1.4.
Review and drill Ch1.

## 04 Sep: Autonomous systems and applications section 2.1

Collected Page 41, 1.4: 18, 22, 26
Some more class discussion of 1.5-34.
Due Wed, Page 54, 1.5: 20, 34.
Drill on Section 1.5: Three linear examples: y'+(1/x)y=1, y'+y=x, y'+2y=1.
Drill: classification separable, quadrature, linear.
Drill: Methods for solving first order equations:
1. Linear integrating factor method,
2. Superposition + equilibrium solution for constant-coefficient linear,