# 2250-1 7:30am Lecture Record Week 3 S2012

Last Modified: January 29, 2012, 19:34 MST.    Today: July 18, 2018, 12:34 MDT.

## 23 Jan: Autonomous systems and applications section 2.1

```Drill on 1.5 Problems
There are two special methods for solving y'+py=q
If p,q are constant then use the SUPERPOSITION METHOD
y = y_p + y_h
y_p = an equilibrium solution (set y'=0, solve for y)
y_h = constant divided by the integrating factor
If one of p or q depends on x, then use the STANDARD METHOD
Replace the LHS, which is y'+p(x)y, by the integrating
factor quotient (Wy)'/W, where W=exp(int p(x)dx)) is the
integrating factor.
Cross-multiply by W to clear fractions. Then apply the
```

Slides: Linear integrating factor method (129.8 K, pdf, 03 Mar 2012)
Manuscript: Linear DE part I. Integrating Factor Method (152.7 K, pdf, 07 Aug 2009)
```General Verhulst DE
Solving y'=(a-by)y by a substitution
Let u=y/(a-by).
Then substitution into the DE gives u'=au
Solve u'=au to get u=u0 exp(ax).
Back-substitute u(x) into u=y/(a-by), then solve for y.
Solving y'=(a-by)y by partial fractions
Divide the DE by (a-by)y
Find the constants in the partial fractions on the left.
a y0
y(x) = --------------------------
b y0 + (a - b y0) exp(-ax)
where y0=y(0)=initial population size.
```

## 24 Jan: Autonomous Differential Equations and Phase Diagrams. Sections 2.1, 2.2

```Lecture on 2.2:
Theory of autonomous DE y'=f(y)
Picard's theorem and non-crossing of solutions.
Direction fields and translation of solutions
No direction field is needed to draw solution curves
We throw out the threaded solution rule used in chapter 1,
replace it by two rules from calculus and a theorem:
1. If y'(x)>0, then y(x) increases.
2. If y'(x)<0, then y(x) decreases.
THEOREM. For y'=f(y), a threaded solution starting with
y'(0)>0 must satisfy y'(x)>0 for x>0. A similar result
holds for y'(0)<0.
Definition: phase line diagram, phase diagram,
Calculus tools: f'(x) pos/neg ==> increasing/decreasing
DE tool: solutions don't cross
Maple tools for production work.
Stability theory of autonomous DE y'=f(y)
Stability of equilibrium solutions.
Stable and unstable classification of equilibrium solutions.
funnel, spout, node,
How to construct Phase line diagrams
How to make a phase diagram graphic
Inventing a graph window
Invention of the grid points
Using the phase line diagram to make the graphic
calculus tools
DE tools
```

html: Problem notes S2012 (5.2 K, html, 08 Apr 2012)
```Examples and Applications
Growth-Decay model y'=ky and its algebraic model y=y(0)exp(kx).
Pharmokinetics of drug transport [ibuprofen]
Pollution models.
Three lake pollution model [Erie, Huron, Ontario].
Brine tanks.
One-tank model.
Two-tank and three-tank models.
Recycled brine tanks and limits of chapter 1 methods.
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 the shortcut for
homogeneous DE y'+py=0.
Separation of variables
The equation y'=7y(y-13), y(0)=17
F(x) = 7, G(y) = y(y-13)
Separated form y'/G(y) = F(x)
Answer check using the Verhulst solution
P(t) = aP_0/(bP_0 + (a-b P_0)exp(-at))
Separation of variables details.
Review of partial fractions.
Partial fraction details for 1/((u(u-13)) = A/u + B/(u-13)
```

## Jan 25 and 27: Newton Kinematic Models. Projectiles. Problem. Section 2.3.

```Drill and Review
Phase diagram for y'=y^2(y^2-4)
Phase line diagram
Labels: stable, unstable, funnel, spout, node

Phase line diagrams.
Phase diagram.
```
```Newton's force and friction models
Isaac Newton ascent and descent kinematic models.
Free fall with no air resistance F=0.
Linear air resistance models F=kx'.
Non-linear air resistance models F=k|x'|^2.
```

The tennis ball problem. Does it take longer to rise or longer to fall?
Slides: Newton kinematics with air resistance. Projectiles. (137.1 K, pdf, 03 Mar 2012)
Jules Verne problem. A rocket from the earth to the moon.
Slides: Jules Verne Problem (127.3 K, pdf, 03 Mar 2012)
Reading assignment: proofs of 2.3 theorems in the textbook and derivation of details for the rise and fall equations with air resistance.
Problem notes for 2.3-10. 2.3-20 (2.8 K, txt, 28 Jan 2012)
```  Partial fractions.
DEFINITION: partial fraction=constant/polynomial with exactly one root
THEOREM: P(x)/q(x) = a sum of partial fractions
Finding the coefficients.
Method of sampling
clear fractions, substitute samples, solve for A,B, ...
Method of atoms
clear fractions, multiply out and match powers, solve for A,B,...
Heaviside's cover-up method
partially clear fraction, substitute root, find one constant
Separation of variable solutions with partial fractions.
Exercise solutions to the problems due in 2.1.
```

## Jan 26: Patrick

```Exam 1 review, questions and examples on exam problems 1,2,3,4,5.
Sample Exam: Exam 1 key from F2010. See also S2010, exam 1.
Lecture on midterm 1 problems 4,5. Lecture on 2.2-10,18.
Maple Lab 2, problem 1 details [maple L2.1].
```