# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

2 Pages

### Week08

Course: MATH 2373, Fall 2008
School: Minnesota
Rating:

Word Count: 425

#### Document Preview

2373 Math Week 8 Toews 1 Reduction of Order y (x) + p(x)y (x) + q(x)y(x) = f (x). Consider the general second order linear non-homogenous equation (1) (2) If q 0, this becomes y (x) + p(x)y (x) = f (x), for which we can make the substitution u = y to get u (x) + p(x)u(x) = f (x). (3) Equation (3) is rst order linear in u, and can be solved. Once u is known, y can found as the integral of u. In the...

Register Now

#### Unformatted Document Excerpt

Coursehero >> Minnesota >> Minnesota >> MATH 2373

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
2373 Math Week 8 Toews 1 Reduction of Order y (x) + p(x)y (x) + q(x)y(x) = f (x). Consider the general second order linear non-homogenous equation (1) (2) If q 0, this becomes y (x) + p(x)y (x) = f (x), for which we can make the substitution u = y to get u (x) + p(x)u(x) = f (x). (3) Equation (3) is rst order linear in u, and can be solved. Once u is known, y can found as the integral of u. In the general case, this substitution will not work. However, if we happen to know one solution y1 to the homogenous equation (1), a second solution can be assumed to have the form y2 (x) = u(x)y1 (x). (4) Expanding y2 and y2 using the product rule, substituting in (1), and using the fact that y1 is a solution of the homogenous equation, we get u + u p(x) + 2 f y1 = . y1 y1 (5) Equation (5) is rst order linear, and can be solved for u. Once u is known, y2 is given by (4). 2 Resonance The method of undetermined coecients amounted to guessing an appropriate form of the particular solution for the equation y + ay + by = f (6) where f is either trigonometric, or polynomial, exponential (or some combination of these.) The notes for Week 6 outline the appropriate guesses whenever f is not also a solution to the homogenous equation. If f is a solution to the 1 homogenous equation, the usual guess should be multiplied by a factor of x. A proof of this fact can be developed using the method of Reduction of Order. Recall that spring systems and LRC problems involve an equation of the form (6), where the coecients are positive. If the spring system has no damping or the LRC system has no resistence, then the second term in (6) is zero and the solution to the homogenous equation is a f...

Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

Minnesota - MATH - 2373
Minnesota - CHEM - 4101
First name Christine Sara Rita Beryl Miseung Katrina Robert Megan Derek Aimee Rith Katie Abbey Laura Joshua Observations:Topic Anti-anxiety medication safety Stents with drugs to prevent resenosis Rocket fuel spills in water contaminate food meat a
Minnesota - CAMM - 0010
Research Bibliography Second Languages and Cultures Education Prepared for CI 8631-8632Allright, R. L. (1988). Observation in the language classroom. New York: Longman. Bernard, H. R. (1988). Research methods in cultural anthropology. Newbury Park
Oklahoma State - DOCUMENT - 2304
Oklahoma Cooperative Extension ServiceEPP-7168Plant Galls Caused by Insects and MitesTom RoyerExtension EntomologistDon C. ArnoldSurvey EntomologistOklahoma Cooperative Extension Fact Sheets are also available on our website at: http:/osuf
Oklahoma State - DOCUMENT - 2137
Oklahoma Cooperative Extension ServiceANSI-3655Pasture Cooling for Bred SowsRaymond L. Huhnke William G. LuceExtension Agricultural EngineerExtension Swine SpecialistOklahoma Cooperative Extension Fact Sheets are also available on our websi
Oklahoma State - DOCUMENT - 5308
Oklahoma Cooperative Extension ServiceEPP-7323Managing Storm-Damaged TreesDamon L. Smith Eric J. RebekAssistant Professor/State Specialist, Horticultural Crops PathologyOklahoma Cooperative Extension Fact Sheets are also available on our webs
Oklahoma State - DOCUMENT - 2609
Oklahoma Cooperative Extension ServiceAGEC-227Adjusting and Setting Up Mechanical Dockage TestersPhil KenkelExtension EconomistKim AndersonExtension EconomistOklahoma Cooperative Extension Fact Sheets are also available on our website at:
Oklahoma State - DOCUMENT - 2047
Oklahoma Cooperative Extension ServiceANSI-3400Home Slaughtering and Processing of BeefHaroldR.HedlickandWilliamC.ShingelDepartmentofFoodScienceandNutritionMauriceAlexanderDepartmentofAnimalHusbandry,CollegeofAgricultureOklahomaCooperativ
Minnesota - HOTMETRICS - 08
HotMetrics 2008 Session I: Systems Services, and EnergyScribe: Brian L. Mark, George Mason University1Image Management in a Virtualized Data CenterPresenter: Tingxi Tan, University of Calgary Discussant: Question: How do you specify the perfor
Minnesota - ROZAI - 001
Gathering Materials1. HTML-Building for WebCTIn WebCT Campus Edition (CE), you will almost always need to convert your text documents into HTML documents.1 It's not necessary to know HTML in order to use WebCT, but it's sometimes helpful to have a
Minnesota - ROZAI - 001
GRAD 8101 ASSIGNMENTSAUTOBIOGRAPHY OF A LEARNERAn autobiography of a learner tells the story of your experiences with teaching and learning and explores the impact they have had on you as a learner and a teacher. The assignment serves three purpos
Minnesota - NORD - 0380
POL3235W Nordquist Summer 2008 Writing a 12-15 page paper (four-credit option) Step One: Generating a Topic The four-credit option requires you to write a longer, 12-15 page paper for the course. Such papers, to be well-conceived and well-written, ta
Minnesota - ZIEF - 0002
'EPsy 8262Regression &amp; The General Linear Model Lab #2\$DEVELOPING A DEEP UNDERSTANDING OF STATISTICAL INFERENCEIn your rst homework assignment, you investigated the relationship between the graduation rates of Black and White college football
Minnesota - MODULE - 9
Poetry UnitTOM DESHOTELS / SPRING 2004This unit is inspired by Sheridan Blaus The Literature Workshop: Teaching Texts and Their Readers, Bruce Emras Coming of Age: Literature About Youth and Adolescence, Katie Schultzs poetry unit, and Jennifer Vi
Minnesota - MODULE - 12
Film analysis using formalist lensJENNIFER LARSON / MAY 12, 2004PurposesThis unit is for 11th grade Advanced Placement English studying AP Language and Composition. It will be the first unit of the year. Students will already have read The Things
Minnesota - MODULE - 12
Literary Film Adaptations: Chaucer ExtravaganzaAMY GUSTAFSON &amp; KATHY CONNORS / MAY 12, 2004Unit Context Level: High School o Because of the Literature we picked, this would be for a British Literature Course. The literature used would all have a
Minnesota - MODULE - 10
A bright idea: bring biases out of the darkHEATHER JOHNSON / JULY 18, 2002 The mission of this unit is to identify the many levels of bias in the media, primarily in the news. The scope of bias can be carried throughout every type of media and thi
Minnesota - MODULE - 11
Video Curriculum: Documentary Production and AnalysisNORINA BECK / SEPTEMBER 10, 2002Subject: Community and HomeThis curriculum seeks to incorporate the techniques of video production with an understanding of how these techniques shape our interp
Minnesota - MILLE - 066
Geol 4110 Take home Final Reports - EARTH SCIENCE TEACHING UNIT DUE FRIDAY, MAY 114/05/07Objective: For your final exam, you will be asked to develop a 10- (individually) to 15-hour (as a pair) teaching unit on an Earth Science theme and grade le
Minnesota - SHIN - 0104
Introduction (1) A Case from China: Fertility PolicyLocal Society in Modern State:The Imperative of the Dual-rule GameRule 1: Chinese National Fertility Policy 1n nOne Couple, One Child Relatively Universal Rule Local Knowledge and Practice M
Minnesota - CLAR - 0514
Chapter 15Reading and Writing Social ResearchHow to Read Social Research Begin with the abstract. Skim the piece. Read the conclusion to get a good sense of what it is about. Form questions as you read and take notes. Abstract A summary of a
Minnesota - PSY - 2007
Andy Hertel Introduction to Social PsychologyTodayPsychology 3201: Introduction to Social PsychologyElliott Hall Room N119 Monday, January 29 4:00-6:00PM 4:00-Social cognitionSchemasEffects Nature TypesSocial cognitionWe are interdependen
Minnesota - PSY - 3201
Andy Hertel Introduction to Social PsychologyTodayPsychology 3201: Introduction to Social PsychologyElliott Hall Room N119 Monday, January 29 4:00-6:00PM 4:00-Social cognitionSchemasEffects Nature TypesSocial cognitionWe are interdependen
Oklahoma State - MATH - 4023
52. 0 si Z fo ytilanidrac eht os dna ,N Z .,suhT .noitcejib a si 2 dd o si n fi 1 + n n Z N : f 2 neve si n fi n pam eht taht yfirev ot ysae si ti nehT . srebmun larutan fo tes eht ,N yb dna ,sregetni fo tes eht eht ,Z yb etoned ew lausu sA . 0
Oklahoma State - MATH - 3013
LECTURE 6Inverses of Square Matrices1. Introduction To motivate our discussion of matrix inverses, let me recall the solution of a linear equation in one variable: (6.1) ax = bThis is achieved simply by multiplying both sides by a1 . Put another
Oklahoma State - MATH - 4023
03.0 &gt; n dna ,0 = na ,sregetni l la er a 0a , . . . , 1-na , na stneicffieoc eht erehw 0 = 0a + x 1a + + 1-nx 1-na + nx na )1.9( n oitauqe laimonyl op a sefisit as fi fi rebmun ciarbegla na del lac si rebmun A .1.9 noitinif eD .rehtruf neve dlef
Minnesota - EE - 5239
EE5239Prof. T. LuoSolution of Homework #2(1.2.1) f (x, y) = 3x2 + y 4 , x0 = Therefore, f (x0 ) = 19 and f (x, y)|x0 = (a) For steepest descent algorithm, we have d0 = - f (x0 ) = Step size can be found by the Armijo rule: f( or f( 1 -2 + (.5)m
Minnesota - EE - 8950
EE 5581 Information Theory &amp; Coding Homework 1 Due: Friday, Feb. 8, 5:00 PMProf. N. Jindal Feb. 1, 20081. (a) Write expressions for the mutual information achieved (per real-dimension) by 2-PAM and 4-PAM in an AWGN channel. Recall that an M-PAM c
Minnesota - EE - 5505
EE 5505 Wireless Communication Homework 6 Due: Tuesday, April 11, 5 PM 1. 11.5 (a,b) 2. 11.12 3. 12.7 4. 12.10 5. 12.11Prof. N. Jindal March 26, 20066. You are designing an OFDM system over a passband bandwidth of 10 MHz at a carrier frequency of
Minnesota - SW - 5
Liquid Crystalline Elastomers as Artificial MusclesP.E. CladisAdvanced Liquid Crystal Tech. Summit, NJ 07902-1314May 24, 2005Advanced Liquid Crystal Technologies, Inc.1Stress-Strain in Smectic A SLCEsStrasbourg ILCC 1998 Nicole Assfalg
Minnesota - SW - 5
Photo-Actuation in Liquid Crystal Elastomers Mark Warner Cavendish Laboratory, Cambridge.Nematic elastomers. changes in liquid crystal ordering gives polymer shape change. heat or light-induced distortions of rubber. Actuation thermo-actuation: 4
Minnesota - SW - 5
Minnesota - SW - 5
Robert B. MeyerOctober 1st, 2004Complex Fluids GroupBrandeis UniversityThermo/Electro-Mechanical Instabilitiesin Conned Samples of Nematic Gels Robert B. Meyer &amp; Guangnan MengComplex Fluids Group, Martin Fisher School of Physics Brandeis Un
Minnesota - T - 1
Knot theory and proteinsIsabel K. Darcy University of Iowa www.math.uiowa.edu/~idarcyFrom: A deeply knotted protein structure and how it might fold, William R. Taylor, Nature 406, 916919(24 August 2000)Intricate Knots in Proteins: Function and E
Minnesota - UBICOMP - 2003
Multimodal Interactive User Interfaces for Mobile Multi-Device EnvironmentsRobbie Schaefer, Wolfgang Mueller robbie@c-lab.de, wolfgang@c-lab.de Paderborn University/C-LAB Fuerstenallee 11 Paderborn, GermanyAbstractPortable devices come with diffe
Minnesota - UBICOMP - 2003
Multi-Device, Ambient-Aware Information DeliveryBernd Mrohs and Christian RckFraunhofer Institute FOKUS Competence Center for Open Communication Systems {mrohs,raeck}@fokus.fraunhofer.deAbstract. Today, a huge number of wearable devices enable us
Minnesota - GCLUTO - 1
gCLUTO DocumentationVersion 1.2Matt Rasmussen, Mark Newman, George Karypis University of Minnesota. Copyright 2004 rasm0146@umn.edu, karypis@cs.umn.edu October 7, 2004gCLUTO DocumentationTable of Contents1 Introduction..1 1.1 What is gCLUTO..1
Minnesota - GCLUTO - 1
gCLUTO DocumentationVersion 1.1Matt Rasmussen, Mark Newman, George Karypis University of Minnesota. Copyright 2003 rasm0146@umn.edu, karypis@cs.umn.edu June 20, 2004gCLUTO DocumentationTable of Contents1 Introduction..1 1.1 What is gCLUTO..1 1
Minnesota - CLASS - 1
Assignment # 3Selected SolutionsProblem 3.4 Prove that in a group, |a| = |a1 |. Solution 1 Proof by contradiction. Recall that |a| = n means that n is the smallest number such that an = e. Now suppose that |a| = n, |a1 | = m and n = m. We can suppo
Minnesota - CLASS - 1
Assignment # 5-Selected Solutions Problem 4.38 Solution From Cayley table you observe that the line by element 16 is 4, 8, 12, 16, hence e = 16. Now, 42 = 16, so |4| = 2 and 4 is not a generator. On the other hand, 81 = 8, 82 = (64) = 4, 83 = (512) =
Minnesota - MATH - 5378
PROBLEM SET 11: SOLUTIONS6.3.2 d/ds(V ) = V + V = 0 alpha + V 0 = 0. Similarly, d/ds(V V ) = 0 and d/ds( ) = 0. This means V , |V |, and | | are constant, and therefore, cos = V /(|V | |) is also constant. Thus so is . For the ang
Minnesota - MATH - 5378
Math 5378, Differential Geometry Solutions to practice questions for Test 1 1. Find a parametrized curve whose trace is the set of points (x, y) in R2 with xy = 1, x &gt; 0. Solution: One possible solution is (t) = (t, 1/t) for t (0, ). Remember to spe
Minnesota - MEREV - 001
Solutions to Homework 6FM 5001 Preparation for Financial Mathematics1-1 Determine if (1, 2, 3, 4, -44) is in the span of (1, 0, 3, 6, -2), (-2, 2, -4, -4, 10), (4, 4, 17, 18, 1), (-3, 6, -4, -9, -10). The last four vectors are the rows of the matri
Minnesota - MEREV - 001
Solutions to Homework 10FM 5001 Preparation for Financial Mathematics1. Let 1 0 2 M := 4 1 4 . 4 0 5 1-1. Find the eigenvalues of M . The eigenvalues of M satisfy the equation det(M I) = 0. 1 0 2 1 4 det(M I) = det 4 4 0 5 = (1 ) det
Minnesota - MATH - 5011
FM 5011Mathematical Background for FinanceFall 2008For # 1,2,3 Dene a PCRV (piecewise continuous random variable) 2, if 4, if Y () = 3, if 15, if (10/16)-1-1a Find the center of mass of the distribution of Y .Y : [0, 1] R by 0.00 0.
Minnesota - MATH - 5001
FM 5001Preparation for Financial MathematicsFall 20081. Denitions. Complete the following sentences. (a) (5 points) A set in Rn is closed if . . . Solution: . . . it contains all its boundary points. (b) (5 points) A function: f : Rn R is comp
Wisc Stevens Point - TSCHA - 816
Assignment Sheet Singing assignment (Students must have this assignment completed by - one week prior to instrumental rehearsal of this piece) = WTS 1 For the most part, movement one is an atonal work. Adler reserves movement two as the pieces tonal
Minnesota - MATH - 5011
FM 5011Mathematical Background for FinanceFall 2008Let Let Let LetZ := -1 be a standard normal random variable. X := 2Z - 1 be a normal variable with SD = 2 and mean = -1. F (x) = P r[X x] be the CDF of X. Y := F X be the grade of X.(10/3
Minnesota - CSCI - 1901
CSci 1901 Fall 2006 Midterm Exam 1 (100 points)This is a closed book exam. Do not use notes, books, computers, calculators, nor any other materials other than a pen and this exam booklet in completing this exam. You are encouraged to show your work
Minnesota - CSCI - 1901
CSci 1901 Fall 2007 Midterm Exam 2 (100 points)This is a 50-minute closed book exam. Do not use notes, books, computers, calculators, nor any other materials other than a pen and this exam booklet in completing this exam. You are encouraged to show
Minnesota - CSCI - 2011
CSCI2011: Discrete Structures of Computer Science (Fall 2007)Foundations (I):Propositional Logic and Predicate Logic(Chapter 1.1-1.4, Rosen 6th Edition)Rui KuangDepartment of Computer Science and Engineering University of Minnesota kuang@cs.um
Minnesota - CSCI - 5801
Lecture 8 - Requirements: What are they really?Fall 2007Fall 2007CSci 5801 - Dr. Mats Heimdahl1The WRSPM ModelMainly Jackson and ZaveObjective For TodayUnderstanding what requirements really are Reference ModelsFall 2007CSci 5801 -
Minnesota - CSCI - 5801
Lecture 12: ModelsThe Importance of Good RequirementsConcept Formation??Most problems here!Requirements Specificationhttp:/www.umsec.umn.eduDesign(Requirements) ModelingGuest Lecture: Ajitha Rajan?Implementation2How we Develop
Minnesota - CSCI - 5107
Minnesota - CSCI - 5801
Software Engineering ICSci 5801 Fall 2007 Midterm 2 Suggested SolutionsThis is a 75-minute exam. It is a closed book test. On all essay type questions you will receive points based on the quality of the answer - not the quantity. Be to the point
Minnesota - CSCI - 5801
Software Engineering ICSci 5801 - Fall 2004 Midterm 1Question 2 - 10 Points.Why are incremental process models considered by many to be the best approach to software development in a modern context? Proposed Solution: Because time lines for the de
Minnesota - CSCI - 5107
Minnesota - CSCI - 2031
CSci 2031 Fall 2007 Assignment 5 Key (1) The scale matrix is [6, 3, 3] throughout the procedure. The first pivot row is therefore row 2, and after the first iteration of the k loop, the index vector is [2, 1, 3], and the coefficient matrix is3/2 -
Minnesota - CSCI - 2011
CSci 2011 ! Discrete Mathematics!Lecture 2 ch1.1! Propositional Logic!Fall 2008 Yongdae KimCSci 2011 Fall 2008Recap! http:/www-users.itlabs.umn.edu/classes/Fall-2008/csci2011/ E-mail policy Include [2011] in the subject of your e-mail CC TA
Minnesota - CSCI - 5211
Data Communication and Computer Networks Assignment V1. a. Describe the differences between a bridge and a router.Sol:Bridge Operate on layer 2 Flat address space Broadcast to every port when next hop port information is unknown. Connects two p
Minnesota - CSCI - 5481
CSCI 5481: Computational Techniques in Genomics (Fall 2008)Sequence AlignmentRui KuangDepartment of Computer Science and Engineering University of Minnesota (The slides are modified from lecture notes by Serafim Batzoglou and bioalgorithms.info)