Edwards-Penney, sections 6.2, 6.3, 7.3, 7.4, 9.1, 9.2 The textbook topics, definitions and theorems

Edwards-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)

Edwards-Penney 9.1,9.2,9.3,9.4 (12.1 K, txt, 18 Dec 2013)

Exam 2 ReviewSample exam 2, Problem typesDiagonalization TheoryIn 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 independentDrill Problems1. Problem: Given P and D, find A in the relation AP=PD. 2. Problem: Given Fourier's model, find A. 3. Problem: Given A, find Fourier's model. 4. Problem: Given A, find all eigenpairs. 5. Problem: Given A, find packages P and D such that AP=PD. 6. Problem: Give an example of a matrix A which has no Fourier's model. 7. Problem: Give an example of a matrix A which is not diagonalizable. 8. Problem: Given 2 eigenpairs, find the 2x2 matrix A.Cayley-Hamilton topics, Section 6.3.Power MethodComputing 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.Google AlgorithmLawrence Page's pagerank algorithm, google web page rankings.

Methods to solve dynamical systemsConsider the 2x2 system x'=x-5y, y'=x-y, x(0)=1, y(0)=2. Cayley-Hamilton-Ziebur method. Laplace resolvent. Eigenanalysis method. Exponential matrix using maple Putzer's method to compute the exponential matrix [slides, not studied in 2250] Spectral methods [ch8; not studied in 2250]

Survey of Methods for solving a 2x2 dynamical system1. 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 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 7 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.

Exam 2 ReviewSample exam 2, Chapter 6 problems 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 and Final exam: Second shifting theorem in Laplace theory.Second Order SystemsHow 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 EquationsThe model u'' = Ax + F(t)Characteristic Equation for a Second Order SystemEuler's Substitution: u = exp(lambda t) v u = solution vector in u'' = Au v = unknown vector, to be determined lambda = scale factor, a number real or complex Substitute Euler's expression into u'' = Au, solve for lambda and v. Then the pair (lambda,v) determines a solution u of u'' = AuCoupled Spring-Mass System. Problem 7.4-6A:=matrix([[-6,4],[2,-4]]); eigenvals(A); lambda1= -2, lambda2= -8Ziebur's Method and Euler's Characteristic EquationWhy Ziebur's method applies and the First Order System w'=Bw 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 atomsMaple routines for second orderde1:=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 section 7.4u(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 7.4-24Cayley-Hamilton-Ziebur method Laplace Resolvent method for second order Eigenanalysis method section 7.4

Some Deprecated Topics 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.

Extra Credit Maple Project: Tacoma narrows. Explore an alternative explanation for what caused the bridge to fail, based on the hanging cables.: Maple Lab 9. Tacoma Narrows (0.0 K, pdf, 31 Dec 1969)MAPLEExtra Credit Maple Project: Earthquakes. Explore a 5-story or 7-story building and the resonant frequencies of oscillation of the building which might make it destruct during an earthquake. See Edwards-Penney, application section in 7.4.: Maple Lab 8. Earthquakes (0.0 K, pdf, 31 Dec 1969)MAPLE

Dynamical Systems TopicsEquilibria. Stability. Instability. Asymptotic stability. Classification of equilibria for u'=Au when det(A) is not zero, for the 2x2 case. Impact of Cayley-Hamilton-Ziebur on classification

Slides on Dynamical Systems: Systems theory and examples (730.9 K, pdf, 10 Apr 2014)Manuscript: Laplace second order systems, spring-mass,boxcars, earthquakes (288.1 K, pdf, 03 Mar 2012)Slides: Introduction to dynamical systems (145.4 K, pdf, 05 Apr 2015)Slides: Phase Portraits for dynamical systems (221.3 K, pdf, 05 Apr 2015)Slides: Stability for dynamical systems (154.4 K, pdf, 05 Apr 2015)Slides: Nonlinear classification spiral, node, center, saddle (98.0 K, pdf, 05 Apr 2015)Slides: Matrix Exponential, Putzer Formula, Variation Parameters (130.1 K, pdf, 03 Mar 2012)Slides

References for Eigenanalysis and Systems of Differential Equations.: 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)Slides: Systems of DE examples and theory (730.9 K, pdf, 10 Apr 2014)Manuscript: Home heating, attic, main floor, basement (99.3 K, pdf, 09 Apr 2014)Slides: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008)Text: History of telecom companies (1.9 K, txt, 03 Apr 2013)Text

Systems of Differential Equations applications: Cable hoist example (73.2 K, pdf, 21 Aug 2008)Slides: Sliding plates example (105.8 K, pdf, 21 Aug 2008)SlidesExtra Credit Maple Project: Tacoma narrows. Explore an alternative explanation for what caused the bridge to fail, based on the hanging cables.: Maple Lab 9. Tacoma Narrows (0.0 K, pdf, 31 Dec 1969)MAPLELaplace theory references: 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)Slides: Maple Lab 7. Laplace applications (0.0 K, pdf, 31 Dec 1969)MAPLE: DE systems, examples, theory (730.9 K, pdf, 10 Apr 2014)Manuscript: 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)Slides: Heaviside's method 2008 (352.3 K, pdf, 07 Jan 2014)Manuscript: Laplace theory 2008 (497.3 K, pdf, 19 Mar 2014)Manuscript: Ch10 Laplace solutions 10.1 to 10.4 (1068.7 K, pdf, 28 Nov 2010)Transparencies: Laplace theory problem notes (17.7 K, txt, 17 Mar 2014)Text: Final exam study guide (8.3 K, txt, 06 Jan 2015)Text