Notes 16 Linearly Independence and Wronskians
|
|
- Edwin Carson
- 5 years ago
- Views:
Transcription
1 ECE 6382 Fall 2017 David R. Jackson otes 16 Linearly Independence and Wronskians otes are from D. R. Wilton, Dept. of ECE 1
2 Linear Independence Definition: { φ } A set of functions ( x), n = 1, 2,, is linearly independent if the c n = 0, n. n only solution to the set of homogeneous equations is n= 1 c φ ( x) 0 n n A set of functions that is not linearl y independent is said to be linearly dependent. 2
3 Linear Independence (cont.) Assume linear dependence: n= 1 c φ ( x) 0 n n cn 0 Assume cp 0 Then c c c c c φ = φ φ φ φ φ p 1 p+ 1 p p 1 p 1 c + p c p c p c p c p or φ = aφ + a φ + a φ + a φ + a φ p 1 2 p 1 p 1 p+ 1 p+ 1 Conclusion: One of the functions can be written as a combination of the others. 3
4 Linear Independence (cont.) Examples : (sin x,cos x) are linearly independent 2 3 (,, ) x x x 2 2 (,, 2 ) x x x + x are linearly independent are linearly dependent 4
5 Linear Independence (cont.) ote: The classification of linear independence depends in general on the interval for x that is being considered. Example : f ( x) 1, x 0 cos x, x 0 ( ) g x 2, x 0 x + 2, x 0 These two functions are linearly independent on any interval (a, b) that contains positive x values. These two functions are linearly dependent on any interval that contains only negative x values. 5
6 Wronskians and Linear Independence The Wronskian allows us to test for linear independence on some interval. φ1 φ2 φ φ1 φ2 φ φ 1 φ 2 φ φ 1 φ 2 φ W( x) det = φ φ φ φ φ φ ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) We look at whether or not the Wronskian vanishes identically over an interval. Josef Hoëné-Wronski 6
7 Wronskians and Linear Independence (cont.) Assume linear dependence: For some value of p φ = aφ + a φ + a φ + a φ + a φ : p 1 2 p 1 p 1 p+ 1 p+ 1 Then, taking derivatives, we h ave : φ = aφ + a φ + a φ + a φ + a φ p 1 2 p 1 p 1 p+ 1 p+ 1 φ = aφ + a φ + a φ + a φ + a φ p 1 2 p 1 p 1 p+ 1 p+ 1 φ = aφ + a φ + a φ + a φ + a φ p 1 2 p 1 p 1 p+ 1 p+ 1 ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) φ = aφ + a φ + a φ + a φ + a φ p 1 2 p 1 p 1 p+ 1 p+ 1 The pth column of the Wronskian matrix is thus a combination of the other columns, so the determiant is zero. φ φ φ φ φ φ = = ( ) 0 φ φ φ ( 1) ( 1) ( 1) Hence Linear dependence ( ) 0 7
8 Wronskians and Linear Independence (cont.) Interestingly, ( ) 0 Linear dependence Example (Peano): ( ) 2 ( x, xx) 0, but we have linear independence on any interval ( ab, ) that includes the origin! However, if the functions are analytic, then ( ) 0 linear dependence (proof by Peano). Giuseppe Peano 8
9 Wronskians and Linear Independence (cont.) Summary For arbitrary functions: Linear dependence ( ) 0 For analytic functions: Linear dependence ( ) 0 9
10 Wronskians and Linear Independence (cont.) Example Show that φ ( x) = e, φ ( x) = e, φ ( x) = sinkx are linearly dependent. ikx ikx 3 φ φ φ ikx ikx e e sin kx ikx ikx φ φ φ = ike ike k coskx = 0 φ φ φ 2 ikx 2 ikx 2 ke ke k kx 2 ( k ) (Last row is first row! ) sin ote: The three functions are analytic, so W 0 is equivalent to linear dependence. We can also show that any pair of solutions are linearly independent : ikx ikx φ ( x) = e, φ ( x) = e, φ ( x) = sin kx, φ ( x) = coskx
11 Wronskians for SOLDE*s Assume y ( x), y ( x) are two independent solutions of Hence Also, we have Hence y = y + p( x) y + q( x) y = 0 W( x) = = yy yy 2 1 y 1 y 2 W = yy y y + yy yy 2 1 yy 2 1 ( ) y2( py 1 qy1 ) = y py qy 2 ( Wx) ( ) = p y y y y = pw W = W W = W d dx d dx 2 1 px ( ) ( ln W( x) ) ln ( ) = px ( ) *SOLDE: Second-Order Linear Differential Equation 11
12 Wronskians for SOLDEs (cont.) From the last slide : d dx ( Wx) ln ( ) = px ( ) Integrate from x= ato x= x: x W( x) ln W ( x) ln W ( a) = ln = p( x) dx W( a) a p( x) dx W( x) = W( a) e a > 0 x The Wronskian either vanishes for all x( W( a) = 0 ) or for no x ( W( a) 0)! If the two solutions to a SOLDE are linearly independent in one region, they must be linearly independent everywhere. 12
Homework #6 Solutions
Problems Section.1: 6, 4, 40, 46 Section.:, 8, 10, 14, 18, 4, 0 Homework #6 Solutions.1.6. Determine whether the functions f (x) = cos x + sin x and g(x) = cos x sin x are linearly dependent or linearly
More informationMATH 3321 Sample Questions for Exam 3. 3y y, C = Perform the indicated operations, if possible: (a) AC (b) AB (c) B + AC (d) CBA
MATH 33 Sample Questions for Exam 3. Find x and y so that x 4 3 5x 3y + y = 5 5. x = 3/7, y = 49/7. Let A = 3 4, B = 3 5, C = 3 Perform the indicated operations, if possible: a AC b AB c B + AC d CBA AB
More informationMA Ordinary Differential Equations
MA 108 - Ordinary Differential Equations Santanu Dey Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 76 dey@math.iitb.ac.in March 21, 2014 Outline of the lecture Second
More information2. Second-order Linear Ordinary Differential Equations
Advanced Engineering Mathematics 2. Second-order Linear ODEs 1 2. Second-order Linear Ordinary Differential Equations 2.1 Homogeneous linear ODEs 2.2 Homogeneous linear ODEs with constant coefficients
More information1 Solution to Homework 4
Solution to Homework Section. 5. The characteristic equation is r r + = (r )(r ) = 0 r = or r =. y(t) = c e t + c e t y = c e t + c e t. y(0) =, y (0) = c + c =, c + c = c =, c =. To find the maximum value
More informationREFERENCE: CROFT & DAVISON CHAPTER 20 BLOCKS 1-3
IV ORDINARY DIFFERENTIAL EQUATIONS REFERENCE: CROFT & DAVISON CHAPTER 0 BLOCKS 1-3 INTRODUCTION AND TERMINOLOGY INTRODUCTION A differential equation (d.e.) e) is an equation involving an unknown function
More informationMath 322. Spring 2015 Review Problems for Midterm 2
Linear Algebra: Topic: Linear Independence of vectors. Question. Math 3. Spring Review Problems for Midterm Explain why if A is not square, then either the row vectors or the column vectors of A are linearly
More informationMA4001 Engineering Mathematics 1 Lecture 15 Mean Value Theorem Increasing and Decreasing Functions Higher Order Derivatives Implicit Differentiation
MA4001 Engineering Mathematics 1 Lecture 15 Mean Value Theorem Increasing and Decreasing Functions Higher Order Derivatives Implicit Differentiation Dr. Sarah Mitchell Autumn 2014 Rolle s Theorem Theorem
More informationAPPLIED MATHEMATICS. Part 1: Ordinary Differential Equations. Wu-ting Tsai
APPLIED MATHEMATICS Part 1: Ordinary Differential Equations Contents 1 First Order Differential Equations 3 1.1 Basic Concepts and Ideas................... 4 1.2 Separable Differential Equations................
More informationMath 334 A1 Homework 3 (Due Nov. 5 5pm)
Math 334 A1 Homework 3 Due Nov. 5 5pm No Advanced or Challenge problems will appear in homeworks. Basic Problems Problem 1. 4.1 11 Verify that the given functions are solutions of the differential equation,
More informationOWELL WEEKLY JOURNAL
Y \»< - } Y Y Y & #»»» q ] q»»»>) & - - - } ) x ( - { Y» & ( x - (» & )< - Y X - & Q Q» 3 - x Q Y 6 \Y > Y Y X 3 3-9 33 x - - / - -»- --
More informationSeries Solution of Linear Ordinary Differential Equations
Series Solution of Linear Ordinary Differential Equations Department of Mathematics IIT Guwahati Aim: To study methods for determining series expansions for solutions to linear ODE with variable coefficients.
More informationDIFFERENTIAL EQUATIONS
Mr. Isaac Akpor Adjei (MSc. Mathematics, MSc. Biostats) isaac.adjei@gmail.com April 7, 2017 ORDINARY In many physical situation, equation arise which involve differential coefficients. For example: 1 The
More informationMB4018 Differential equations
MB4018 Differential equations Part II http://www.staff.ul.ie/natalia/mb4018.html Prof. Natalia Kopteva Spring 2015 MB4018 (Spring 2015) Differential equations Part II 0 / 69 Section 1 Second-Order Linear
More informationGeorgia Tech PHYS 6124 Mathematical Methods of Physics I
Georgia Tech PHYS 612 Mathematical Methods of Physics I Instructor: Predrag Cvitanović Fall semester 2012 Homework Set #5 due October 2, 2012 == show all your work for maximum credit, == put labels, title,
More informationLinear DifferentiaL Equation
Linear DifferentiaL Equation Massoud Malek The set F of all complex-valued functions is known to be a vector space of infinite dimension. Solutions to any linear differential equations, form a subspace
More information17.2 Nonhomogeneous Linear Equations. 27 September 2007
17.2 Nonhomogeneous Linear Equations 27 September 2007 Nonhomogeneous Linear Equations The differential equation to be studied is of the form ay (x) + by (x) + cy(x) = G(x) (1) where a 0, b, c are given
More information= 2e t e 2t + ( e 2t )e 3t = 2e t e t = e t. Math 20D Final Review
Math D Final Review. Solve the differential equation in two ways, first using variation of parameters and then using undetermined coefficients: Corresponding homogenous equation: with characteristic equation
More informationThe Dirac δ-function
The Dirac δ-function Elias Kiritsis Contents 1 Definition 2 2 δ as a limit of functions 3 3 Relation to plane waves 5 4 Fourier integrals 8 5 Fourier series on the half-line 9 6 Gaussian integrals 11 Bibliography
More informationMATH 260 Homework assignment 8 March 21, px 2yq dy. (c) Where C is the part of the line y x, parametrized any way you like.
MATH 26 Homework assignment 8 March 2, 23. Evaluate the line integral x 2 y dx px 2yq dy (a) Where is the part of the parabola y x 2 from p, q to p, q, parametrized as x t, y t 2 for t. (b) Where is the
More informationu u + 4u = 2 cos(3t), u(0) = 1, u (0) = 2
MATH HOMEWORK #6 PART A SOLUTIONS Problem 7..5. Transform the given initial value problem into an initial value problem for two first order equations. u + 4 u + 4u cost, u0, u 0 Solution. Let x u and x
More informationOrdinary differential equations Notes for FYS3140
Ordinary differential equations Notes for FYS3140 Susanne Viefers, Dept of Physics, University of Oslo April 4, 2018 Abstract Ordinary differential equations show up in many places in physics, and these
More information" W I T H M I A L I O E T O W A R D istolste A N D O H A P l t T Y F O B, A I j L. ; " * Jm MVERSEO IT.
P Y V V 9 G G G -PP - P V P- P P G P -- P P P Y Y? P P < PG! P3 ZZ P? P? G X VP P P X G - V G & X V P P P V P» Y & V Q V V Y G G G? Y P P Y P V3»! V G G G G G # G G G - G V- G - +- - G G - G - G - - G
More informationDifferential Equations
Differential Equations Problem Sheet 1 3 rd November 2011 First-Order Ordinary Differential Equations 1. Find the general solutions of the following separable differential equations. Which equations are
More information17.8 Nonhomogeneous Linear Equations We now consider the problem of solving the nonhomogeneous second-order differential
ADAMS: Calculus: a Complete Course, 4th Edition. Chapter 17 page 1016 colour black August 15, 2002 1016 CHAPTER 17 Ordinary Differential Equations 17.8 Nonhomogeneous Linear Equations We now consider the
More informationLecture 6: Spanning Set & Linear Independency
Lecture 6: Elif Tan Ankara University Elif Tan (Ankara University) Lecture 6 / 0 Definition (Linear Combination) Let v, v 2,..., v k be vectors in (V,, ) a vector space. A vector v V is called a linear
More informationProblem 1 (Equations with the dependent variable missing) By means of the substitutions. v = dy dt, dv
V Problem 1 (Equations with the dependent variable missing) By means of the substitutions v = dy dt, dv dt = d2 y dt 2 solve the following second-order differential equations 1. t 2 d2 y dt + 2tdy 1 =
More informationMath 113 Solutions: Homework 8. November 28, 2007
Math 113 Solutions: Homework 8 November 28, 27 3) Define c j = ja j, d j = 1 j b j, for j = 1, 2,, n Let c = c 1 c 2 c n and d = vectors in R n Then the Cauchy-Schwarz inequality on Euclidean n-space gives
More informationA First Course in Elementary Differential Equations: Problems and Solutions. Marcel B. Finan Arkansas Tech University c All Rights Reserved
A First Course in Elementary Differential Equations: Problems and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved 1 4 The Method of Variation of Parameters Problem 4.1 Solve y
More informationLinear Independence. MATH 322, Linear Algebra I. J. Robert Buchanan. Spring Department of Mathematics
Linear Independence MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Given a set of vectors {v 1, v 2,..., v r } and another vector v span{v 1, v 2,...,
More informationECE Spring Prof. David R. Jackson ECE Dept. Notes 26
ECE 6345 Spring 05 Prof. David R. Jackson ECE Dept. Notes 6 Overview In this set of notes we use the spectral-domain method to find the mutual impedance between two rectangular patch ennas. z Geometry
More informationECE Spring Prof. David R. Jackson ECE Dept. Notes 15
ECE 6341 Spring 216 Prof. David R. Jackson ECE Dept. Notes 15 1 Arbitrary Line Current TM : A (, ) Introduce Fourier Transform: I I + ( k ) jk = I e d x y 1 I = I ( k ) jk e dk 2π 2 Arbitrary Line Current
More informationReview for Exam 2. Review for Exam 2.
Review for Exam 2. 5 or 6 problems. No multiple choice questions. No notes, no books, no calculators. Problems similar to homeworks. Exam covers: Regular-singular points (5.5). Euler differential equation
More informationكلية العلوم قسم الرياضيات المعادالت التفاضلية العادية
الجامعة اإلسالمية كلية العلوم غزة قسم الرياضيات المعادالت التفاضلية العادية Elementary differential equations and boundary value problems المحاضرون أ.د. رائد صالحة د. فاتن أبو شوقة 1 3 4 5 6 بسم هللا
More informationChapter 4. Higher-Order Differential Equations
Chapter 4 Higher-Order Differential Equations i THEOREM 4.1.1 (Existence of a Unique Solution) Let a n (x), a n,, a, a 0 (x) and g(x) be continuous on an interval I and let a n (x) 0 for every x in this
More informationChapter 3 Higher Order Linear ODEs
Chapter 3 Higher Order Linear ODEs Advanced Engineering Mathematics Wei-Ta Chu National Chung Cheng University wtchu@cs.ccu.edu.tw 1 2 3.1 Homogeneous Linear ODEs 3 Homogeneous Linear ODEs An ODE is of
More informationODE. Philippe Rukimbira. Department of Mathematics Florida International University PR (FIU) MAP / 92
ODE Philippe Rukimbira Department of Mathematics Florida International University PR (FIU) MAP 2302 1 / 92 4.4 The method of Variation of parameters 1. Second order differential equations (Normalized,
More information6. Linear Differential Equations of the Second Order
September 26, 2012 6-1 6. Linear Differential Equations of the Second Order A differential equation of the form L(y) = g is called linear if L is a linear operator and g = g(t) is continuous. The most
More informationThe Theory of Second Order Linear Differential Equations 1 Michael C. Sullivan Math Department Southern Illinois University
The Theory of Second Order Linear Differential Equations 1 Michael C. Sullivan Math Department Southern Illinois University These notes are intended as a supplement to section 3.2 of the textbook Elementary
More informationLecture 31. Basic Theory of First Order Linear Systems
Math 245 - Mathematics of Physics and Engineering I Lecture 31. Basic Theory of First Order Linear Systems April 4, 2012 Konstantin Zuev (USC) Math 245, Lecture 31 April 4, 2012 1 / 10 Agenda Existence
More informationLOWELL WEEKLY JOURNAL
Y G y G Y 87 y Y 8 Y - $ X ; ; y y q 8 y $8 $ $ $ G 8 q < 8 6 4 y 8 7 4 8 8 < < y 6 $ q - - y G y G - Y y y 8 y y y Y Y 7-7- G - y y y ) y - y y y y - - y - y 87 7-7- G G < G y G y y 6 X y G y y y 87 G
More informationChapter 3. Reading assignment: In this chapter we will cover Sections dx 1 + a 0(x)y(x) = g(x). (1)
Chapter 3 3 Introduction Reading assignment: In this chapter we will cover Sections 3.1 3.6. 3.1 Theory of Linear Equations Recall that an nth order Linear ODE is an equation that can be written in the
More informationElementary Differential Equations, Section 2 Prof. Loftin: Practice Test Problems for Test Find the radius of convergence of the power series
Elementary Differential Equations, Section 2 Prof. Loftin: Practice Test Problems for Test 2 SOLUTIONS 1. Find the radius of convergence of the power series Show your work. x + x2 2 + x3 3 + x4 4 + + xn
More information0.1 Problems to solve
0.1 Problems to solve Homework Set No. NEEP 547 Due September 0, 013 DLH Nonlinear Eqs. reducible to first order: 1. 5pts) Find the general solution to the differential equation: y = [ 1 + y ) ] 3/. 5pts)
More informationLecture notes: Introduction to Partial Differential Equations
Lecture notes: Introduction to Partial Differential Equations Sergei V. Shabanov Department of Mathematics, University of Florida, Gainesville, FL 32611 USA CHAPTER 1 Classification of Partial Differential
More informationChapter 13: General Solutions to Homogeneous Linear Differential Equations
Worked Solutions 1 Chapter 13: General Solutions to Homogeneous Linear Differential Equations 13.2 a. Verifying that {y 1, y 2 } is a fundamental solution set: We have y 1 (x) = cos(2x) y 1 (x) = 2 sin(2x)
More informationEXERCISE SET 5.1. = (kx + kx + k, ky + ky + k ) = (kx + kx + 1, ky + ky + 1) = ((k + )x + 1, (k + )y + 1)
EXERCISE SET 5. 6. The pair (, 2) is in the set but the pair ( )(, 2) = (, 2) is not because the first component is negative; hence Axiom 6 fails. Axiom 5 also fails. 8. Axioms, 2, 3, 6, 9, and are easily
More informationMath 4263 Homework Set 1
Homework Set 1 1. Solve the following PDE/BVP 2. Solve the following PDE/BVP 2u t + 3u x = 0 u (x, 0) = sin (x) u x + e x u y = 0 u (0, y) = y 2 3. (a) Find the curves γ : t (x (t), y (t)) such that that
More informationCheng Soon Ong & Christian Walder. Canberra February June 2018
Cheng Soon Ong & Christian Walder Research Group and College of Engineering and Computer Science Canberra February June 2018 (Many figures from C. M. Bishop, "Pattern Recognition and ") 1of 89 Part II
More informationHigher Order Linear Equations
C H A P T E R 4 Higher Order Linear Equations 4.1 1. The differential equation is in standard form. Its coefficients, as well as the function g(t) = t, are continuous everywhere. Hence solutions are valid
More informationQuantum Theory. Thornton and Rex, Ch. 6
Quantum Theory Thornton and Rex, Ch. 6 Matter can behave like waves. 1) What is the wave equation? 2) How do we interpret the wave function y(x,t)? Light Waves Plane wave: y(x,t) = A cos(kx-wt) wave (w,k)
More informationNotes 19 Gradient and Laplacian
ECE 3318 Applied Electricity and Magnetism Spring 218 Prof. David R. Jackson Dept. of ECE Notes 19 Gradient and Laplacian 1 Gradient Φ ( x, y, z) =scalar function Φ Φ Φ grad Φ xˆ + yˆ + zˆ x y z We can
More informationLINEAR DIFFERENTIAL EQUATIONS. Theorem 1 (Existence and Uniqueness). [1, NSS, Section 6.1, Theorem 1] 1 Suppose. y(x)
LINEAR DIFFERENTIAL EQUATIONS MINSEON SHIN 1. Existence and Uniqueness Theorem 1 (Existence and Uniqueness). [1, NSS, Section 6.1, Theorem 1] 1 Suppose p 1 (x),..., p n (x) and g(x) are continuous real-valued
More informationPower Series Solutions And Special Functions: Review of Power Series
Power Series Solutions And Special Functions: Review of Power Series Pradeep Boggarapu Department of Mathematics BITS PILANI K K Birla Goa Campus, Goa September, 205 Pradeep Boggarapu (Dept. of Maths)
More informationLinear Algebra (wi1403lr) Lecture no.4
Linear Algebra (wi1403lr) Lecture no.4 EWI / DIAM / Numerical Analysis group Matthias Möller 29/04/2014 M. Möller (EWI/NA group) LA (wi1403lr) 29/04/2014 1 / 28 Review of lecture no.3 1.5 Solution Sets
More informationNATIONAL OPEN UNIVERSITY OF NIGERIA SCHOOL OF SCIENCE AND TECHNOLOGY COURSE CODE: MTH421 COURSE TITLE: ORDINARY DIFFERENTIAL EQUATIONS
MTH 421 NATIONAL OPEN UNIVERSITY OF NIGERIA SCHOOL OF SCIENCE AND TECHNOLOGY COURSE CODE: MTH421 COURSE TITLE: ORDINARY DIFFERENTIAL EQUATIONS MTH 421 ORDINARY DIFFERENTIAL EQUATIONS COURSE WRITER Prof.
More informationLS.5 Theory of Linear Systems
LS.5 Theory of Linear Systems 1. General linear ODE systems and independent solutions. We have studied the homogeneous system of ODE s with constant coefficients, (1) x = Ax, where A is an n n matrix of
More informationdt 2 roots r = 1 and r =,1, thus the solution is a linear combination of e t and e,t. conditions. We havey(0) = c 1 + c 2 =5=4 and dy (0) = c 1 + c
MAE 305 Assignment #3 Solutions Problem 9, Page 8 The characteristic equation for d y,y =0isr, = 0. This has two distinct roots r = and r =,, thus the solution is a linear combination of e t and e,t. That
More informationSeries Solutions of Linear ODEs
Chapter 2 Series Solutions of Linear ODEs This Chapter is concerned with solutions of linear Ordinary Differential Equations (ODE). We will start by reviewing some basic concepts and solution methods for
More information2.3 Linear Equations 69
2.3 Linear Equations 69 2.3 Linear Equations An equation y = fx,y) is called first-order linear or a linear equation provided it can be rewritten in the special form 1) y + px)y = rx) for some functions
More informationEx. 1. Find the general solution for each of the following differential equations:
MATH 261.007 Instr. K. Ciesielski Spring 2010 NAME (print): SAMPLE TEST # 2 Solve the following exercises. Show your work. (No credit will be given for an answer with no supporting work shown.) Ex. 1.
More informationMaths 381 The First Midterm Solutions
Maths 38 The First Midterm Solutions As Given by Peter Denton October 8, 8 # From class, we have that If we make the substitution y = x 3/, we get # Once again, we use the fact that and substitute into
More informationy x 3. Solve each of the given initial value problems. (a) y 0? xy = x, y(0) = We multiply the equation by e?x, and obtain Integrating both sides with
Solutions to the Practice Problems Math 80 Febuary, 004. For each of the following dierential equations, decide whether the given function is a solution. (a) y 0 = (x + )(y? ), y =? +exp(x +x)?exp(x +x)
More informationLecture 9. Scott Pauls 1 4/16/07. Dartmouth College. Math 23, Spring Scott Pauls. Last class. Today s material. Group work.
Lecture 9 1 1 Department of Mathematics Dartmouth College 4/16/07 Outline Repeated Roots Repeated Roots Repeated Roots Material from last class Wronskian: linear independence Constant coeffecient equations:
More informationChapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form
Chapter 5. Linear Algebra A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are real numbers. 1
More information3. Identify and find the general solution of each of the following first order differential equations.
Final Exam MATH 33, Sample Questions. Fall 7. y = Cx 3 3 is the general solution of a differential equation. Find the equation. Answer: y = 3y + 9 xy. y = C x + C x is the general solution of a differential
More informationMath 313 Chapter 5 Review
Math 313 Chapter 5 Review Howard Anton, 9th Edition May 2010 Do NOT write on me! Contents 1 5.1 Real Vector Spaces 2 2 5.2 Subspaces 3 3 5.3 Linear Independence 4 4 5.4 Basis and Dimension 5 5 5.5 Row
More informationA matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and
Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.
More informationNonhomogeneous Equations and Variation of Parameters
Nonhomogeneous Equations Variation of Parameters June 17, 2016 1 Nonhomogeneous Equations 1.1 Review of First Order Equations If we look at a first order homogeneous constant coefficient ordinary differential
More informationStatistical Learning Theory
Statistical Learning Theory Part I : Mathematical Learning Theory (1-8) By Sumio Watanabe, Evaluation : Report Part II : Information Statistical Mechanics (9-15) By Yoshiyuki Kabashima, Evaluation : Report
More informationLinear Equations in Linear Algebra
1 Linear Equations in Linear Algebra 1.5 SOLUTION SETS OF LINEAR SYSTEMS HOMOGENEOUS LINEAR SYSTEMS A system of linear equations is said to be homogeneous if it can be written in the form A x= 0, where
More informationJackson 2.3 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jackson.3 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell PROBLEM: A straight-line charge with constant linear charge λ is located perpendicular to the x-y plane in
More informationA matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and
Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.
More informationHomogeneous Linear Systems and Their General Solutions
37 Homogeneous Linear Systems and Their General Solutions We are now going to restrict our attention further to the standard first-order systems of differential equations that are linear, with particular
More informationAnswer Key b c d e. 14. b c d e. 15. a b c e. 16. a b c e. 17. a b c d. 18. a b c e. 19. a b d e. 20. a b c e. 21. a c d e. 22.
Math 20580 Answer Key 1 Your Name: Final Exam May 8, 2007 Instructor s name: Record your answers to the multiple choice problems by placing an through one letter for each problem on this answer sheet.
More informationQuantum Theory. Thornton and Rex, Ch. 6
Quantum Theory Thornton and Rex, Ch. 6 Matter can behave like waves. 1) What is the wave equation? 2) How do we interpret the wave function y(x,t)? Light Waves Plane wave: y(x,t) = A cos(kx-wt) wave (w,k)
More information5.4 Independence, Span and Basis
318 54 Independence, Span and Basis The technical topics of independence, dependence and span apply to the study of Euclidean spaces R 2, R 3,, R n and also to the continuous function space C(E), the space
More informationMath 250B Final Exam Review Session Spring 2015 SOLUTIONS
Math 5B Final Exam Review Session Spring 5 SOLUTIONS Problem Solve x x + y + 54te 3t and y x + 4y + 9e 3t λ SOLUTION: We have det(a λi) if and only if if and 4 λ only if λ 3λ This means that the eigenvalues
More informationReview. To solve ay + by + cy = 0 we consider the characteristic equation. aλ 2 + bλ + c = 0.
Review To solve ay + by + cy = 0 we consider the characteristic equation aλ 2 + bλ + c = 0. If λ is a solution of the characteristic equation then e λt is a solution of the differential equation. if there
More information374: Solving Homogeneous Linear Differential Equations with Constant Coefficients
374: Solving Homogeneous Linear Differential Equations with Constant Coefficients Enter this Clear command to make sure all variables we may use are cleared out of memory. In[1]:= Clearx, y, c1, c2, c3,
More informationA( x) B( x) C( x) y( x) 0, A( x) 0
3.1 Lexicon Revisited The nonhomogeneous nd Order ODE has the form: d y dy A( x) B( x) C( x) y( x) F( x), A( x) dx dx The homogeneous nd Order ODE has the form: d y dy A( x) B( x) C( x) y( x), A( x) dx
More informationSection 4.7: Variable-Coefficient Equations
Cauchy-Euler Equations Section 4.7: Variable-Coefficient Equations Before concluding our study of second-order linear DE s, let us summarize what we ve done. In Sections 4.2 and 4.3 we showed how to find
More informationPower series solutions for 2nd order linear ODE s (not necessarily with constant coefficients) a n z n. n=0
Lecture 22 Power series solutions for 2nd order linear ODE s (not necessarily with constant coefficients) Recall a few facts about power series: a n z n This series in z is centered at z 0. Here z can
More informationQMI Lesson 19: Integration by Substitution, Definite Integral, and Area Under Curve
QMI Lesson 19: Integration by Substitution, Definite Integral, and Area Under Curve C C Moxley Samford University Brock School of Business Substitution Rule The following rules arise from the chain rule
More informationIIT JEE Maths Paper 2
IIT JEE - 009 Maths Paper A. Question paper format: 1. The question paper consists of 4 sections.. Section I contains 4 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for
More information(c)
1. Find the reduced echelon form of the matrix 1 1 5 1 8 5. 1 1 1 (a) 3 1 3 0 1 3 1 (b) 0 0 1 (c) 3 0 0 1 0 (d) 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 (e) 1 0 5 0 0 1 3 0 0 0 0 Solution. 1 1 1 1 1 1 1 1
More informationLECTURE 14: REGULAR SINGULAR POINTS, EULER EQUATIONS
LECTURE 14: REGULAR SINGULAR POINTS, EULER EQUATIONS 1. Regular Singular Points During the past few lectures, we have been focusing on second order linear ODEs of the form y + p(x)y + q(x)y = g(x). Particularly,
More informationMATH 391 Test 1 Fall, (1) (12 points each)compute the general solution of each of the following differential equations: = 4x 2y.
MATH 391 Test 1 Fall, 2018 (1) (12 points each)compute the general solution of each of the following differential equations: (a) (b) x dy dx + xy = x2 + y. (x + y) dy dx = 4x 2y. (c) yy + (y ) 2 = 0 (y
More information1D Wave PDE. Introduction to Partial Differential Equations part of EM, Scalar and Vector Fields module (PHY2064) Richard Sear.
Introduction to Partial Differential Equations part of EM, Scalar and Vector Fields module (PHY2064) November 12, 2018 Wave equation in one dimension This lecture Wave PDE in 1D Method of Separation of
More informationKeywords: Acinonyx jubatus/cheetah/development/diet/hand raising/health/kitten/medication
L V. A W P. Ky: Ayx j//m// ///m A: A y m "My" W P 1986. S y m y y. y mm m. A 6.5 m My.. A { A N D R A S D C T A A T ' } T A K P L A N T A T { - A C A S S T 0 R Y y m T ' 1986' W P - + ' m y, m T y. j-
More informationNote : This document might take a little longer time to print. more exam papers at : more exam papers at : more exam papers at : more exam papers at : more exam papers at : more exam papers at : more
More informationLecture 4. 1 de Broglie wavelength and Galilean transformations 1. 2 Phase and Group Velocities 4. 3 Choosing the wavefunction for a free particle 6
Lecture 4 B. Zwiebach February 18, 2016 Contents 1 de Broglie wavelength and Galilean transformations 1 2 Phase and Group Velocities 4 3 Choosing the wavefunction for a free particle 6 1 de Broglie wavelength
More informationCHAPTER 2. Techniques for Solving. Second Order Linear. Homogeneous ODE s
A SERIES OF CLASS NOTES FOR 005-006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES A COLLECTION OF HANDOUTS ON SCALAR LINEAR ORDINARY DIFFERENTIAL
More informationSection 8.2 : Homogeneous Linear Systems
Section 8.2 : Homogeneous Linear Systems Review: Eigenvalues and Eigenvectors Let A be an n n matrix with constant real components a ij. An eigenvector of A is a nonzero n 1 column vector v such that Av
More informationSign the pledge. On my honor, I have neither given nor received unauthorized aid on this Exam : 11. a b c d e. 1. a b c d e. 2.
Math 258 Name: Final Exam Instructor: May 7, 2 Section: Calculators are NOT allowed. Do not remove this answer page you will return the whole exam. You will be allowed 2 hours to do the test. You may leave
More informationNon-homogeneous equations (Sect. 3.6).
Non-homogeneous equations (Sect. 3.6). We study: y + p(t) y + q(t) y = f (t). Method of variation of parameters. Using the method in an example. The proof of the variation of parameter method. Using the
More informationHandbook of Ordinary Differential Equations
Handbook of Ordinary Differential Equations Mark Sullivan July, 28 i Contents Preliminaries. Why bother?...............................2 What s so ordinary about ordinary differential equations?......
More informationChapter 8 Indeterminate Forms and Improper Integrals Math Class Notes
Chapter 8 Indeterminate Forms and Improper Integrals Math 1220-004 Class Notes Section 8.1: Indeterminate Forms of Type 0 0 Fact: The it of quotient is equal to the quotient of the its. (book page 68)
More informationLinear Algebra 1 Exam 2 Solutions 7/14/3
Linear Algebra 1 Exam Solutions 7/14/3 Question 1 The line L has the symmetric equation: x 1 = y + 3 The line M has the parametric equation: = z 4. [x, y, z] = [ 4, 10, 5] + s[10, 7, ]. The line N is perpendicular
More informationECONOMICS 207 SPRING 2006 LABORATORY EXERCISE 5 KEY. 8 = 10(5x 2) = 9(3x + 8), x 50x 20 = 27x x = 92 x = 4. 8x 2 22x + 15 = 0 (2x 3)(4x 5) = 0
ECONOMICS 07 SPRING 006 LABORATORY EXERCISE 5 KEY Problem. Solve the following equations for x. a 5x 3x + 8 = 9 0 5x 3x + 8 9 8 = 0(5x ) = 9(3x + 8), x 0 3 50x 0 = 7x + 7 3x = 9 x = 4 b 8x x + 5 = 0 8x
More information