# 2280 12:55pm Lectures Week 11 S2015

Last Modified: April 05, 2015, 19:22 MDT.    Today: December 16, 2018, 06:27 MST.
``` Edwards-Penney, sections 4.1, 4.2, 5.1 to 5.6
The textbook topics, definitions and theoremsEdwards-Penney 4.1 to 4.3 (10.4 K, txt, 05 Jan 2015)Edwards-Penney 5.1 to 5.6 (22.6 K, txt, 04 Jan 2015)```

### Week 11: Sections 5.1, 5.2, 5.3, 5.4

#### Monday and Tuesday (after Spring Break): Systems of differential equations, 5.1, 5.2, 5.3

```Review of Eigenanalysis
Eigenvalue
Eigenvector
Eigenpair
Fourier's Model
Diagonalizable matrix

Main theorem for solving u'=Au by eigenanalysis
Example 1. Consider the 2x2 system
x'=x+3y, y'=2y, x(0)=1, y(0)=-1.
Method 1: Linear integrating factor method for triangular systems
Method 2: Cayley-Hamilton-Ziebur, textbook shortcuts applied to 2x2
Method 3: Eigenanalysis
Method 4: Laplace resolvent
Method 5: Exponential matrix, maple or Putzer's method
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: Matrix Exponential, Putzer Formula, Variation Parameters (123.5 K, pdf, 29 Mar 2015)```
```Applications
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

Methods to solve dynamical systems
Example 2. Consider the 2x2 system
x'=x-5y, y'=x-y, x(0)=1, y(0)=2.
Cayley-Hamilton-Ziebur method.
Textbook shortcut preferred, section 4.2 example 1
Laplace resolvent.
Eigenanalysis method.
Exponential matrix using maple
Putzer's method to compute the exponential matrix [slides, not in the textbook]
```
``` Survey of Methods for solving a 2x2 dynamical system
1. Cayley-Hamilton-Ziebur method for u'=Au
Solution: u(t)=(atom_1)vec(d_1)+ (atom_2)vec(d_2)
Atoms: They are constructed by Euler's theorem from roots of det(A-rI)=0
THEOREM. Vectors vec(d_1),vec(d_2) are found from the equation
[d1 | d2]=[u(0) | Au(0)](W(0)^T)^(-1)
where W(t) is the Wronskian matrix of the two atoms.
2. Laplace resolvent L(u)=(s I - A)^(-1) u(0)
See slides for details about the resolvent equation.
3. Eigenanalysis  u(t) = exp(lambda_1 t) v1 + exp(lambda_2 t) v2
See chapter 5 in Edwards-Penney for examples and details.
This method fails when matrix A is not diagonalizable.
EXAMPLE. Solve a homogeneous system u'=Au, u(0)=vector([1,2]),
A=matrix([[2,3],[0,4]]) using
Zeibur's method, Laplace resolvent and eigenanalysis.

Next week: 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? (124.0 K, pdf, 14 Nov 2007) Slides: Advanced topics in Linear Algebra, Cayley-Hamilton, Jordan form (332.3 K, pdf, 05 Apr 2015) 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)   Slides: Introduction to dynamical systems (145.4 K, pdf, 05 Apr 2015)   Slides: Matrix Exponential, Putzer Formula, Variation Parameters (123.5 K, pdf, 29 Mar 2015)
```

#### Wednesday and Friday: Eigenanalysis; Second Order Systems. Sections 5.3, 5.4

```   Eigenvalues
A 4x4 matrix.
Block determinant theorem.
Eigenvectors for a 4x4.
B:=matrix([[5,0,0,0],[0,5,0,0],[0,0,0,3],[0,0,-3,0]]);
lambda=5,5,3i,-3i
v1=[1,0,0,0], v2=[0,1,0,0], v3=[0,0,i,-1], v4=[0,0,i,1]
One panel for lambda=5
First frame is A-5I with 0 appended
Find rref
Apply last frame algorithm
Scalar general solution
Take partials on t1, t2 to find v1,v2
Eigenpairs are (5,v1), (5,v2)
One panel for lambda=3i
Same outline as lambda=5
Get one eigenpair (3i,v3)
Other eigenpair=(-3i,v4) where v4 is the conjugate of v3.

Second Order Systems
How to convert mx''+cx'+kx=F0 cos (omega t) into a
dynamical system  u'=Au+F(t).
Electrical systems u'=Au+E(t) from LRC circuit equations.
Electrical systems of order two: networks
Mechanical systems of order two: coupled systems
Second order systems u''=Au+F
Examples are railway cars, earthquakes,
vibrations of multi- component systems,
electrical networks.
Second Order Vector-Matrix Differential EquationsSlides: Laplace second order systems (288.1 K, pdf, 03 Mar 2012)  The model u'' = Ax + F(t)

Coupled Spring-Mass System. Problem 5.3-6
A:=matrix([[-6,4],[2,-4]]); eigenvals(A);
lambda1= -2, lambda2= -8
Ziebur's Method
roots for Ziebur's theorem are plus or minus sqrt(lambda)
Roots = sqrt(2)i,  sqrt(8)i, -sqrt(2)i, -sqrt(8)i
Atoms = cos (sqrt(2)t), sin(sqrt(2)t), cos(sqrt(8)t), sin(sqrt(8)t)
Vector x(t) = vector linear combination of the above 4 atoms
Maple routines for second order
de1:=diff(x(t),t,t)=-6*x(t)+4*y(t); de2:=diff(y(t),t,t)=2*x(t)-4*y(t);
dsolve({de1,de2},{x(t),y(t)});
x(t) = _C1*sin(sqrt(2)*t)+_C2*cos(sqrt(2)*t)+_C3*sin(2*sqrt(2)*t)+_C4*cos(2*sqrt(2)*t),
y(t) = _C1*sin(sqrt(2)*t)+_C2*cos(sqrt(2)*t)-(1/2)*_C3*sin(2*sqrt(2)*t)-(1/2)*_C4*cos(2*sqrt(2)*t)}
Eigenanalysis method Ch5
u(t) = (a1 cos(sqrt(2)t) + b1 sin(sqrt(2)t)) v1 + (a2 cos(sqrt(8)t) + b2 sin(sqrt(8)t)) v2
where (-2,v1), (-8,v2) are the eigenpairs of A.  The two vector terms in u(t) are called
the natural modes of oscillation. The natural frequencies are sqrt(2), sqrt(8).
Eigenanalysis of A gives v1=[1,1], v2=[2,-1].
Railway cars. Problem 5.3-24
Cayley-Hamilton-Ziebur method
Laplace Resolvent method for second order
Eigenanalysis method section 5.3
```
```Matrix Exponential TopicsSlides: Matrix Exponential, Putzer Formula, Variation Parameters (123.5 K, pdf, 29 Mar 2015)
Fundamental matrix Phi(t)
Matrix Variation of Parameters
Definition of the exponential matrix exp(At)
Putzer's method for the 2x2 matrix exponential.
Solution of u'=Au is: u(t) = exp(A t)u(0)
THEOREM: exp(A t) = r1(t) I + r2(t) (A-lambda_1 I),
Lambda Symbols: lambda_1 and lambda_2 are the roots of det(A-lambda I)=0.
The DE System:
r1'(t) = lambda_1 r1(t),         r1(0)=0,
r2'(t) = lambda_2 r2(t) + r1(t), r2(0)=0
See the slides and manuscript on systems for proofs and details.
THEOREM. The formula can be used as
e^{r1 t} - e^{r2 t}
e^{At} = e^{r1 t} I  +  ------------------- (A-r1 I)
r1 - r2
where r1=lambda_1, r2=lambda_2 are the eigenvalues of A.

EXAMPLE. Solve a homogeneous system u'=Au, u(0)=vector([1,2]),
A=matrix([[2,3],[0,4]]) using the matrix exponential,
Zeibur's method, Laplace resolvent and eigenanalysis.
EXAMPLE. Solve a non-homogeneous system u'=Au+F(t), u(0)=vector([0,0]),
A=matrix([[2,3],[0,4]]), F(t)=vector([3,1]) using variation
of parameters.
```