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

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.

- 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.
- Basic theory: 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.

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].

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].

- Venn diagram of classes Q, S, L.
- Examples of various types.
- Test for quadrature (f_y=0)
- 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.

- References for linear DE:

- More about problem 1.5-34
: Problem notes on 1.5-34 (1.9 K, txt, 19 Jul 2009)**Text** - The expected model is x'=1/4-x/16, x(0)=20, using units of millions of cubic feet.
- The answer is x(t)=4+16 exp(-t/16).
- 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].

- Growth-Decay model y'=ky and its algebraic model y=y(0)exp(kx).

Pharmokinetics for drug transport [ibuprofen], brine tanks, pollution models. - One-tank model. Two-tank and three-tank models.
- Recycled brine tanks and non-solvability by chapter 1 methods.
- Three lake pollution model [Erie, Huron, Ontario].
- Linear cascades and how to solve them.
- Method 1: Linear integrating factor method.
- Method 2: Superposition and equilibrium solutions for constant-coefficient y'+py=q. Uses a shortcut for growth-Decay DE y'+py=0

**References for linear applications**: Applications of linear DE (374.2 K, pdf, 28 Jul 2009)**Manuscript**: Brink tanks (62.9 K, pdf, 30 Nov 2009)**Slides**: Home heating (73.8 K, pdf, 30 Nov 2009)**Slides**

Present problems 2, 3 of the midterm 1 sample [S2009 midterm 1 key].**02-03 Sep**: Fusi and Richins

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.: 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)**Answer Key**

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:- Linear integrating factor method,
- Superposition + equilibrium solution for constant-coefficient linear,
- Quadrature method,
- Variables Separable method, includes equilibrium solutions from G(y)=0 and non-equilibrium solutions from G(y) nonzero. factor method,

- Theory of autonomous DE y'=f(y),
- Phase diagrams without a direction field drawing.

**Lecture on 2.1, 2.2:**- References for 2.1, 2.2:

: Autonomous DE (69.9 K, pdf, 03 Sep 2009)**Slides**: Verhulst logistic equation (115.5 K, pdf, 02 Oct 2009)**Manuscript**: Phase Line and Bifurcation Diagrams. Includes Stability, Funnel, Spout, and bifurcation (227.4 K, pdf, 07 Sep 2009)**Manuscript**: ch2 sections 1,2,3: 2.1-6,16,38, 2.2-4,10, 2.3-9,27+Escape velocity (357.6 K, pdf, 29 Jan 2006)**Transparencies**: ch2 DEplot maple example 1 for exercises 2.2, 2.3 (0.7 K, txt, 07 Sep 2009)**Text**: ch2 DEplot maple example 2 for exercises 2.2, 2.3 (0.7 K, txt, 07 Sep 2009)**Text**