# 2250-1 7:30am Lecture Record Week 15 S2012

Last Modified: April 23, 2012, 05:07 MDT.    Today: July 17, 2018, 17:24 MDT.

## Thu Apr 26 Special Final Exam Review

Web 1250, 9-11am. The final exam sample with alternate problems was published 21 April at the final exam review session.
Text: Final exam study guide (8.2 K, txt, 11 Dec 2011)
Pdf: Final exam sample 2012 with alternate problems, questions only (224.4 K, pdf, 23 Apr 2012)
Pdf: Final exam sample 2012 with alternate problems, questions and answers (342.9 K, pdf, 25 Apr 2012)

## Mon Apr 23: Stability. Almost Linear systems. Phase Diagram. Section 9.2

```Review of last week's topics
Phase diagram.
Stability and the three pictures: Node, Center, Spiral
Detecting stability and instability for u'=Au at x=y=0:
Main Theorem: Re(lambda)<0 ==> asym. stability.
Stable center picture. Definition of stability.
Stability at t=-infinity classifies Unstable solutions.
Maple Demonstration
Maple phase diagram tools.
Example
x' = x + y,
y' = 1 - x^2
Classification pictures
Set 1: Stable node, stable center, stable spiral
Set 2: Unstable node, unstable saddle, unstable spiral
How to detect saddle, spiral, node, center in the linear case
using Zeibur's method and examples.
Limitations:
In the case of a node, we cannot sub-classify as improper
or proper using the Zeibur method and examples. The finer
sub-classifications require the exponential matrix e^{At}
or else a synthetic eigenvalue theorem which calculates the
sub-classification.
```
```Spiral, saddle, center, node.
Linearization theory.
Jacobian.

Algebraic Detection of Linear stability for u'=Au:
Rule: det(A) not zero of all classifications!
Re(lambda)<0 ==> asymptotic stability
Re(lambda)=0 and lambda not zero ==> Center picture
Stability at t=-infinity classifies Unstable solutions.
When testing stability, we check t=infinity and t=-infinity.
Nonlinear stability theory u'=f(u)
When the linearized classification and stability transfers to
the nonlinear system.
stability of almost linear [nonlinear] systems,
phase diagrams,
classification of nonlinear systems.
Final exam review started.
Cover today ch8 and some of ch10.
Review packet distributed on the web.
Final exam details
Less contact with ch3, ch4, ch6 due their appearance on
exams 1,2,3.
Since F2008, there are extra chapters 8,9 on the final.
A good sample is the F2010 final exam.
Chapters 5,6,7,10 will undergo changes and spins. For ch10, more
contact with the second shifting theorem and the Dirac Delta. For
ch7-ch8, there are additional methods for solving DE, especially
Cayley-Hamilton-Ziebur, matrix exp(At) and the Laplace resolvent
for first order systems. For ch5, deeper problems on the topics of
variation of parameters and undetermined coefficients, resonance,
and beats.
```

## Tue Apr 24: Nonlinear Stability. Classification. Predator-Prey. Section 9.3

```Nonlinear stability
phase diagrams,
classification.
Predator-Prey systems. How to tell which is the predator and which is
the prey.
Calculations for equilibrium points,
linearization,
classification of equilibria,
impact on the phase diagram.
Using DEtools and DEplot in maple to make phase diagrams.
Exercises 9.1, 9.2.
```

## Wed Apr 25: Nonlinear Mechanical Systems. Section 9.4

```Final exam review continued
Some chapter 8 and chapter 9 problems.
Subspace problems from chapter 4.

Nonlinear mechanical systems.
Hard and soft springs.
Nonlinear pendulum.
Undamped pendulum.
Damped pendulum.
Phase diagrams.
Energy conservation laws and separatrices.
```
Systems of Differential Equations references
Slides: Cable hoist example (73.2 K, pdf, 21 Aug 2008)
Slides: Sliding plates example (105.8 K, pdf, 21 Aug 2008)
Extra Credit Maple Project: Tacoma narrows. Explore an alternative explanation for what caused the bridge to fail, based on the hanging cables.