# 2250 8:05am Lectures Week 13 S2015

Last Modified: April 13, 2015, 05:03 MDT.    Today: December 10, 2018, 09:25 MST.
``` Edwards-Penney, sections 10.5, EPbvp7.6, 6.1, 6.2, 7.1
The textbook topics, definitions and theoremsEdwards-Penney 6.1, 6.2 (7.6 K, txt, 18 Dec 2013)Edwards-Penney 7.1, 7.2, 7.3, 7.4 (25.6 K, txt, 18 Dec 2013)```

### Week 13: 6.2, 7.1, 7.2, 7.3

#### Monday: Eigenanalysis. First Order Systems. Sections 6.1, 6.2, 7.1

```Review of last week
DEF. Eigenpair, eigenvalue, eigenvector
DEF. Fourier's Model. The reason for computing eigenpairs.

THEOREM. The eigenvalues of A are found from the determinant
equation
det(A -lambda I)=0,
which is called the characteristic equation.

THEOREM. The eigenvectors of A are found from the frame sequence which starts
with B=A-lambda I [lambda a root of the characteristic equation, previous Theorem],
ending with last frame rref(B).

The eigenvectors for lambda are the partial derivatives of
the general solution obtained by the Last Frame Algorithm,
with respect to the invented symbols t1, t2, t3, ... These are
Strang's Special solutions for equation Bu=0.

Review: Algebraic Eigenanalysis Section 6.2.
Calculation of eigenpairs to produce Fourier's model.
Connection between Fourier's model and a diagonalizable matrix.
How to find the variables lambda and v in Fourier's model using
determinants and frame sequences.
Solved in class: examples similar to the problems in 6.1 and 6.2.
Web slides and problem notes exist for the 6.1 and 6.2 problems.
Examples where A has an eigenvalue of multiplicity greater than one.

Eigenanalysis Examples
Diagonalization Theory, AP=PD and Fourier's Model
In the case of a 2x2 matrix A,
FOURIER'S MODEL is
A(c1 v1 + c2 v2) = c1(lambda1 v1) + c2(lambda2 v2)
where v1,v2 are a basis for the plane
equivalent to DIAGONALIZATION
AP=PD, where D=diag(lamba1,lambda2), P=augment(v1,v2),
where det(P) is not zero
equivalent to EIGENPAIR EQUATIONS
A(v1)=lambda1 v1, A(v2)=lambda2 v2,
where vectors v1,v2 are independent
```

#### Monday: Sections 6.2,6.3,7.1

```Main Theorem on Fourier's Model
THEOREM. Fourier's model
A(c1 v1 + c2 v2) = c1 (lambda1 v1) + c2 (lambda2 v2)
with v1, v2 a basis of R^2 holds [for all constants c1, c2]
if and only if
the vector-matrix system
A(v1) = lambda1 v1,
A(v2) = lambda2 v2,
has a solution with vectors v1, v2 independent
if and only if
the diagonal matrix D=diag(lambda1,lambda2) and
the augmented matrix P=aug(v1,v2) satisfy
1. det(P) not zero [then v1, v2 are independent]
2. AP=PD
Examples
Given the eigenpairs of A, find A via AP=PD.
Given P, D, then find A.
Given A, then find P, D.
Solving DE System u' = Au by Eigenanalysis
Example: Solving a 2x2 dynamical system
Study of u'=Au, u(0)=vector([1,2]), A=matrix([[2,3],[0,4]]).
Dynamical system scalar form is
x' = 2x + 1y,
y' = 3y,
x(0)=1, y(0)=2.
Find the eigenpairs (2, v1), (3,v2) where v1=vector([1,0])
and v2=vector([1,1]).
THEOREM. The solution of u' = Au in the 2x2 case is
u(t) = c1 exp(lambda1 t) v1 + c2 exp(lambda2 t) v2
APPLICATION:
u(t) = c1 exp(2t) v1 + c2 exp(4t) v2
[ 1 ]            [ 1 ]
u(t) = c1 e^{2t} [   ] + c2 e^4t} [   ]
[ 0 ]            [ 1 ]
which means
x(t) = c1 exp(2t) + 3 c2 exp(4t),
y(t) = 2 c2 exp(4t).

```

#### Wednesday and Friday: Sections 7.1,7.2,7.3

```Section 7.1: Topics from linear systems of differential equations:Manuscript: Systems of Differential Equations, PDF files in 9 sections (0.0 K, 06 Apr 2015)
Systems of two differential equations
Solving a system from Chapter 1 methods: triangular systems, brine cascades
The Laplace resolvent method for systems.
Cramer's Rule,
Matrix inversion methods.
EXAMPLE: Solving a 2x2 dynamical system using Laplace's resolvent method.
Study of u'=Au, u(0)=vector([1,2]), A=matrix([[2,1],[0,3]]).
EXAMPLE: Problem 10.2-16,
This problem is a 3x3 system for x(t), y(t), z(t) solved
by Laplace theory methods. The resolvent formula
(sI - A) L(u(t)) = u(0)
with u(t) the fixed 3-vector with components x(t), y(t), z(t),
amounts to a shortcut to obtain the equations for L(x(t)),
L(y(t)), L(z(t)). After the shortcut is applied, in which
Cramer's Rule is the method of choice, to find the formulas,
there is no further shortcut: we have to find x(t), for example,
by partial fractions and the backward table, followed by Lerch's
theorem.
```
``` EXAMPLE. Recirculating brine tanks
20 x' = -6x + y,
20 y' = 6x - 3y
x(t)=pounds of salt in tank 1 (100 gal)
y(t)=pounds of salt in tank 2 (200 gal)
x(0), y(0) = initial salt amounts in each tank
t=minutes
20=inflow rate=outflow rate
0=inflow salt concentration
EXAMPLE. Solve x'=-2y, y'=x/2.
ANSWER: x(t)=A cos(t) + B sin(t), y(t) = (-A/2) cos(t) +
(B/2) sin(t).
Conversion Methods to Create a First Order System
The position-velocity substitution.
How to convert second order systems.
EXAMPLE. Transform to a first order system
2x'' = -6x + 2y,
y'' = 2x - 2y + 40 sin(3t)
u1' = u2,
u2' = -3u1 + u3, [a division by 2 needed]
u3' = u4,
u4' = 2u1 - 2u3 + 40 sin(3t)
How to convert nth order scalar differential equations.
EXAMPLE. x''' + 2x'' + x = 0
Use u1=x(t), u2=x'(t), u3=x''(t)
Non-homogeneous terms and the vector matrix system
u' = Au + F(t)
Non-linear systems and the vector-matrix system
u' = F(t,u)
Example: The system u'=Au, A=matrix([[2,1],[0,3]]);

Systems of two differential equations
The Laplace resolvent method for systems.
Solving the resolvent equation for L(x), L(y).
Cramer's Rule
Matrix inversion
Elimination
Example: Solving a 2x2 dynamical system
Study of u'=Au, u(0)=vector([1,2]), A=matrix([[2,1],[0,3]]).
Dynamical system scalar form is
x' = 2x + y,
y' = 3y,
x(0)=1, y(0)=2.
The equations for L(x), L(y)
(s-2)L(x)  +  (-1)L(y)=1,
(0)L(x)  + (s-3)L(y)=2
REMARK: Laplace resolvent method shortcut.
How to solve the [resolvent] equations for L(x), L(y).
Cramer's Rule
Matrix inversion
Elimination
delta=(s-2)(s-3), delta1=s-1, delta2=2(s-2)
L(x) = -1/(s-2)+2/(s-3), L(y)=2/(s-3)
Backward table and Lerch's theorem
Answers: x(t) = - e^{2t} + 2 e^{3t},
y(t) = 2 e^{3t}.
Edwards-Penney Shortcut Method in Example 5, 7.1. Uses Chapter 1+5 methods.
This is the Cayley-Hamilton-Ziebur method. See below.
Solve w'+p(t)w=0 as w = constant / integrating factor.
Then  y' -2y=0 ==> y(t) = 2 exp(3t)
Stuff y(t) into the first DE to get the linear DE
x' - 2x = 2 exp(3t)
Superposition: x(t)=x_h(t)+x_p(t),
x_h(t)=c exp(2t),
x_p(t) = d1 exp(t) = 2 exp(3t) by undetermined coeff.
Then x(t)= - exp(2t) + 2 exp(3t).

Cayley-Hamilton Theorem
A matrix satisfies its own characteristic equation.
ILLUSTRATION: det(A-r I)=0 for the previous example
is (2-r)(3-r)=0 or r^2 -5r + 6=0. Then C-H says
A^2 - 5A + 6I = 0.
Cayley-Hamilton-Ziebur Method
ZIEBUR'S LEMMA.
The components of u in u'=Au are linear combinations of
the atoms created by Euler's theorem applied to the
roots of the characteristic equation det(A-rI)=0.
THEOREM. Solve u'=Au without complex numbers or eigenanalysis.
The solution of u'=Au is a linear combination of atoms
times certain constant vectors [not arbitrary vectors].
u(t)=(atom_1)vec(c_1)+ ... + (atom_n)vec(c_n)Slides: Cayley-Hamilton-Ziebur method for solving vector-matrix system u'=Au. (137.7 K, pdf, 28 Mar 2015)
PROBLEM: Solve by Cayley-Hamilton-Ziebur the 2x2 dynamical system
x' = 2x + y,
y' = 3y,
x(0)=1, y(0)=2.
The characteristic equation is (2-lambda)(3-lambda)=0
with roots lambda = 2,3
Euler's theorem implies the atoms are exp(2t), exp(3t).
Ziebur's Theorem says that
u(t) = exp(2t) vec(v_1) + exp(3t) vec(v_2)
where vectors v_1, uv_2 are to be determined from the matrix
A = matrix([[2,1],[0,3]]) and initial conditions x(0)=1, y(0)=2.

CAYLEY-HAMILTON-ZIEBUR ALGORITHM.
To solve for v_1, v_2 in the example, differentiate the
equation u(t) = exp(2t) v_1 + exp(3t) v_2 and set t=0
in both relations. Then u'=Au implies
u_0 =    v_1  +   v_2,
Au_0 = 2 v_1 + 3 v_2.
These equations can be solved by elimination.
v_1 = (3 u_0 -Au_0), v_2 = (Au_0 - 2 u_0)
= vector([-1,0])     = vector([2,2])
Vectors v_1, v_2 are recognized as eigenvectors of A for
lambda=2 and lambda=3, respectively, after studying chapter 6.

ZIEBUR SHORTCUT [Edwards-Penney textbook method, Example 5 in 7.1]
x(t) = k1 exp(2t) + k2 exp(3t).
Use the first DE to solve for y(t):
y(t) = x'(t) - 2x(t)
=  2 k1 exp(2t) + 3 k2 exp(3t)
- 2 k1 exp(2t) - 2 k2 exp(3t))
=   k2 exp(3t)
For example, x(0)=1, y(0)=2 implies k1 and k2 are
defined by
k1 + k2 = 1,
k2 = 2,
which implies k1 = -1, k2 = 2, agreeing with a previous
solution formula.
Cayley-Hamilton topics, Section 6.3.
Cayley-Hamilton Theorem
A matrix satisfies its own characteristic equation.

Proof of the Ziebur Lemma for 2x2 matrices.

Next week: Computing powers of matrices.
Stochastic matrices.
Example of 1984 telecom companies ATT, MCI, SPRINT with discrete
dynamical system u(n+1)=A u(n). Matrix A is stochastic.
EXAMPLE:
[ 6  1  5 ]               [ a(t) ]
10 A = [ 2  7  1 ]        u(t) = [ m(t) ]
[ 2  2  4 ]               [ s(t) ]

Meaning: 60% stay with ATT and 20% switch to MCI, 20% switch to SPRINT.
70% stay with MCI and 20% switch to SPRINT, 10% switch to ATT.
40% stay with SPRINT and 50% switch to ATT, 10% switch to MCI.
Powers of A and the meaning of A^n x_0 for the telecom example.
Lawrence Page's pagerank algorithm, google web page rankings.
Eigenanalysis and powers of A. Text: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008)
Next: Brine tank models.
Recirculating brine tanks.
Pond pollution.
Home heating.
Earthquakes.
Railway cars.
All are 2x2 or 3x3 or nxn system applications that can be solved by Laplace methods.
We investigate 3 fundamental methods: Eigenanalysis, Laplace, Cayley-Hamilton-Ziebur

```
```References for Eigenanalysis and Systems of Differential Equations.
Sildes: Algebraic eigenanalysis (187.6 K, pdf, 03 Mar 2012) Slides: What's eigenanalysis 2008 (174.2 K, pdf, 03 Mar 2012) Slides: What's eigenanalysis, draft 1 (152.2 K, pdf, 01 Apr 2008) Slides: What's eigenanalysis, draft 2 (124.0 K, pdf, 14 Nov 2007) Slides: Cayley-Hamilton-Ziebur method for solving vector-matrix system u'=Au. (137.7 K, pdf, 28 Mar 2015) Slides: Laplace resolvent method (88.1 K, pdf, 03 Mar 2012) Slides: Laplace second order systems (288.1 K, pdf, 03 Mar 2012) Manuscript: Systems of DE examples and theory (730.9 K, pdf, 10 Apr 2014) Slides: Home heating, attic, main floor, basement (99.3 K, pdf, 09 Apr 2014) Text: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008) Text: History of telecom companies (1.9 K, txt, 03 Apr 2013) Slides: Cable hoist example (73.2 K, pdf, 21 Aug 2008) Slides: Sliding plates example (105.8 K, pdf, 21 Aug 2008)
```
```References for Eigenanalysis and Systems of Differential Equations.
Slides: Algebraic eigenanalysis (187.6 K, pdf, 03 Mar 2012) Slides: What's eigenanalysis 2008 (174.2 K, pdf, 03 Mar 2012) Slides: What's eigenanalysis, draft 1 (152.2 K, pdf, 01 Apr 2008) Slides: What's eigenanalysis, draft 2 (124.0 K, pdf, 14 Nov 2007) Slides: Cayley-Hamilton-Ziebur method for solving vector-matrix system u'=Au. (137.7 K, pdf, 28 Mar 2015) Slides: Laplace resolvent method (88.1 K, pdf, 03 Mar 2012) Slides: Laplace second order systems (288.1 K, pdf, 03 Mar 2012) Manuscript: Systems of DE examples and theory (730.9 K, pdf, 10 Apr 2014) Slides: Home heating, attic, main floor, basement (99.3 K, pdf, 09 Apr 2014) Text: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008) Text: History of telecom companies (1.9 K, txt, 03 Apr 2013)

Systems of Differential Equations referencesManuscript: Systems of Differential Equations, PDF files in 9 sections (0.0 K, 06 Apr 2015) 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.
MAPLE: Optional Maple Lab 9. Tacoma Narrows (0.0 K, pdf, 31 Dec 1969)

Laplace theory references
Slides: Laplace and Newton calculus. Photos. (200.2 K, pdf, 03 Mar 2012) Slides: Intro to Laplace theory. Calculus assumed. (144.8 K, pdf, 25 Mar 2015) Slides: Laplace rules (160.3 K, pdf, 03 Mar 2012) Slides: Laplace table proofs (169.6 K, pdf, 03 Mar 2012) Slides: Laplace examples (133.2 K, pdf, 27 Mar 2015) Slides: Piecewise functions and Laplace theory (108.5 K, pdf, 03 Mar 2013) MAPLE: optional Maple Lab 7. Laplace applications (0.0 K, pdf, 31 Dec 1969)
Slides: Laplace resolvent method (88.1 K, pdf, 03 Mar 2012) Slides: Laplace second order systems (288.1 K, pdf, 03 Mar 2012) Slides: Home heating, attic, main floor, basement (99.3 K, pdf, 09 Apr 2014) Slides: Cable hoist example (73.2 K, pdf, 21 Aug 2008) Slides: Sliding plates example (105.8 K, pdf, 21 Aug 2008) Manuscript: Heaviside's method 2008 (352.3 K, pdf, 07 Jan 2014) Manuscript: Laplace theory 2008 (497.3 K, pdf, 19 Mar 2014) Transparencies: Ch10 Laplace solutions 10.1 to 10.4 (1068.7 K, pdf, 28 Nov 2010) Text: Laplace theory problem notes S2013 (17.7 K, txt, 17 Mar 2014) Text: Final exam study guide (8.3 K, txt, 06 Jan 2015)```
```Sample Exam 2 for S2015
PDF: sample exam 2, all problems, answers and solutions. (264.8 K, pdf, 16 Apr 2015)HTML: All sample exams and solution keys. (22.2 K, html, 23 Feb 2015)
```
```EXAM 2 REVIEW:   Sections EPbvp3.7, 5.6, 10.5. EPbvp7.6
Transform Terminology
Convolution theorem and x'' + 4x = cos(t), x(0)=x'(0)=0.
Input
Output
Transfer Function
Circuits EPbvp3.7:
Electrical resonance.
Derivation from mechanical problems 5.6.
THEOREM: omega = 1/sqrt(LC).
Impedance, reactance.
amplitude
Transfer function.
Input and output equation.
Resonance examples
x'' + x = cos(t)
Pure resonance, unbounded solution x(t) = 0.5 t sin(t)
mx'' + cx' + kx = F_0 cos(omega t)
Practical resonance, all solutions bounded, but x(t)
can have extremely large amplitude when omega is tuned
to the frequency omega = sqrt(k/m - c^2/(2m^2))
LQ'' + RQ' + (1/C)Q = E_0 sin(omega t)
Practical resonance, all solutions bounded, but the current
I(t)=dQ/dt can have large amplitude when omega is tuned
to the resonant frequency omega = 1/sqrt(LC).

Resonance Resources
Soldiers marching in cadence, Tacoma narrows bridge, Millennium Bridge
Wine Glass Experiment. Theodore Von Karman and vortex shedding.
Cable model of the Tacoma bridge, year 2000. Resonance explanations.

Extra Credit Maple Project: Tacoma narrows. Explore an alternative
explanation for what caused the bridge to fail, based on the hanging cables.

Chapter 5 referencesSlides: Electrical circuits (112.9 K, pdf, 08 Mar 2014)Slides: Forced damped vibrations (264.0 K, pdf, 08 Mar 2014)Slides: Forced vibrations and resonance (253.0 K, pdf, 08 Mar 2014)Slides: Forced undamped vibrations (214.2 K, pdf, 03 Mar 2012)Slides: Resonance and undetermined coefficients (178.0 K, pdf, 08 Mar 2014)Slides: Unforced vibrations 2008 (647.6 K, pdf, 27 Feb 2014)Slides: Basic undetermined coefficients, draft 5 (156.3 K, pdf, 28 Mar 2013)Slides: Variation of parameters (164.5 K, pdf, 03 Mar 2012)Slides: Resonance and undetermined coefficients (178.0 K, pdf, 08 Mar 2014)
Theory of Practical Resonance
The equation is
mx''+cx'+kx=F_0 cos(omega t)
THEOREM. The limit of x_h(t) is zero at t=infinity
THEOREM. x_p(t) = C(omega) cos(omega t - phi)
C(omega) = F_0/Z, Z^2 = A^2+B^2, A and B are the
undetermined coefficient answers for trial solution
x(t) = A cos(omega t) + B sin(omega t).
THEOREM. The output x(t) = x_h(t) + x_p(t) is graphically
just x_p(t) = C(omega) cos(omega t - phi) for large t.
Therefore, x_p(t) is the OBSERVABLE output.
THEOREM. The amplitude C(omega) is maximized over all possible
input frequencies omega>0 by the single choice
omega = sqrt(k/m - c^2/(2m^2)).
DEFINITION. The practical resonance frequency is the number omega
defined by the above square root expression.

Projection: glass-breaking video. Wine glass experiment. Tacoma narrows.
Video: Wine glass breakage (QuickTime MOV) (96.8 K, mov, 21 Mar 2013)       Video: Wine glass experiment (12mb mpg, 2min) (12493.8 K, mpg, 01 Apr 2008)       Video: Tacoma Narrows Bridge Nov 7, 1940 (18mb mpg, 4min) (18185.8 K, mpg, 01 Apr 2008)
Video: Resonance #17, Wine Glass and Tacoma Narrows (29min Annenburg CPB)

```