Sec. 7.4: Basic Theory of Systems of First Order Linear Equations
|
|
- Mitchell Lester
- 6 years ago
- Views:
Transcription
1 Sec. 7.4: Basic Theory of Systems of First Order Linear Equations MATH 351 California State University, Northridge April 2, 214 MATH 351 (Differential Equations) Sec. 7.4 April 2, / 12
2 System of First Order Linear Equations A system of n first order linear equations: x 1 = p 11(t)x p 1n(t)x n + g 1(t) x 2 = p 21(t)x p 2n(t)x n + g 2(t). x n = p n1(t)x p nn(t)x n + g n(t) To discuss the system (1) most effectively, we write it in matrix notation. That is, x 1(t) g 1(t) p 11(t) p 12(t) p 1n(t) x 2(t) x(t) =., g(t) = g 2(t)., P(t) = p 21(t) p 22(t) p 2n(t)... x n(t) g n(t) p n1(t) p n2(t) p nn(t) Then the system (1) becomes (1) x = P(t)x + g(t) (2) A vector x(t) = φ(t) is said to be a solution of Eq. (2) if its components satisfy the system of equations (1), that is x 1 = φ 1(t), x 2 = φ 2(t),..., x n = φ n(t), (3) which can be viewed as a set of parametric equations in an n-dim space. MATH 351 (Differential Equations) Sec. 7.4 April 2, / 12
3 Recall the Existence and Uniqueness Theorem in Sec. 7.1 Theorem If the functions p 11, p 12,..., p nn, g 1,..., g n are continuous on an open interval I : α < t < β, then there exists a unique solution x 1 = φ 1(t),..., x n = φ n(t) of the system (1) that also satisfies the initial conditions x 1(t ) = x 1, x 2(t ) = x 2,, x n(t ) = x n, (4) where t is any point in I, and x 1, x 2,, x n are any prescribed numbers. Moreover, the solution exists throughout the interval I. Throughout this section, we assume that P and g are continuous on some interval α < t < β; that is, each of the scalar functions p 11, p 12,..., p nn, g 1,..., g n is continuous on α < t < β. MATH 351 (Differential Equations) Sec. 7.4 April 2, / 12
4 Homogeneous Eqn and Nonhomogeneous Eqn If g(t) =, then x = P(t)x is homogeneous. If g(t), then x = P(t)x + g(t) in nonhomogeneous. (Section 7.9) We use the notation x 11(t) x 1k (t) x (1) x 21(t) (t) =.,, x 2k (t) x(k) (t) =., (5) x n1(t) x nk (t) to designate specific solutions of the homogeneous system. NOTE: x ij (t) = x (j) i (t) refers to the ith component of the jth solution x (j) (t). The structure of solutions of system are stated in the following 5 theorems. x = P(t)x (6) MATH 351 (Differential Equations) Sec. 7.4 April 2, / 12
5 Principle of Superposition Theorem If the vector functions x (1) and x (2) are solutions of the system (6), then the linear combination c 1x (1) + c 2x (2) is also a solution for any constants c 1 and c 2. NOTE: This can be proved simply by differentiating c 1x (1) and c 2x (2) and using the fact that x (1) and x (2) satisfy Eq. (6). REMARK: By repeated application of Theorem 7.4.1, we can conclude that if x (1),..., x (k) are solutions of Eq. (3), then x = c 1x (1) (t) + + c k x (k) (t) is also a solution for any constants c 1,..., c k. QUESTION: whether all solutions of Eq. (6) can be found in this way. MATH 351 (Differential Equations) Sec. 7.4 April 2, / 12
6 General Solutions If we properly choose n solutions for the system (6) of n 1st-order eqns, will the linear combinations of these solutions be sufficient? Let x (1),..., x (n) be n solutions of the system (6), and consider the matrix x 11(t) x 1n(t) X(t) =.. (7) x n1(t) x nn(t) Recall from Sec. 7.3 that the columns of X(t) are linearly independent for a given value of t iff det X for that value of t. This determinant is called the Wronskian of the n solutions x (1),..., x (n) (denoted by W [x (1),..., x (n) ]), that is W [x (1),..., x (n) ] = det X(t) (8) Therefore, the solutions x (1),..., x (n) are linearly independent at a point iff W [x (1),..., x (n) ] is not zero there. MATH 351 (Differential Equations) Sec. 7.4 April 2, / 12
7 General Solution and Fundamental Set of Solutions Theorem If the vector functions x (1),..., x (n) are linearly independent solutions of the system (6) for each point in the interval α < t < β, then each solution x = φ(t) of the system (6) can be expressed as φ(t) = c 1x (1) (t) + + c nx (n) (t) (9) in exactly one way. general solution: c 1x (1) (t) + + c nx (n) (t). fundamental set of solutions: any set of solutions of x (1),..., x (n) of is linearly independent Eq. (6) at each point in the interval α < t < β. MATH 351 (Differential Equations) Sec. 7.4 April 2, / 12
8 Proof: We will show that any solution φ(t) of Eq. (6) can be written as for suitable values of c 1,..., c n. φ(t) = c 1x (1) (t) + + c nx (n) (t) MATH 351 (Differential Equations) Sec. 7.4 April 2, / 12
9 Theorem If x (1),..., x (n) are sols of Eq. (6) on the interval α < t < β, then in this interval W [x (1),..., x (n) ] either in identically zero or else never vanishes. Proof: We need to first show that (see Problem 2) dw dt = [p 11(t) + p 22(t) + + p nn(t)]w MATH 351 (Differential Equations) Sec. 7.4 April 2, / 12
10 The existence of fundamental set of solutions Theorem Let 1 1 e (1) =, e (2) =,, e (3) =... 1 further, let x (1),..., x (n) be the solutions of the system (6) that satisfy the initial conditions x (1) (t ) = e (1),..., x (n) (t ) = e (n), (1) respectively, where t is any point in α < t < β. Then x (1),..., x (n) form a fundamental set of solutions of the system (6). Proof: the existence and uniqueness of the solutions x (1),..., x (n) are ensured by Theorem W [x (1),..., x (n) ] = MATH 351 (Differential Equations) Sec. 7.4 April 2, / 12
11 Theorem Consider the system (6) x = P(t)x where each element of P is a real-valued continuous function. If x = u(t) + iv(t) is a complex-valued solution of Eq. (6), then its real part u(t) and its imaginary part v(t) are also solutions of this equation. Proof: MATH 351 (Differential Equations) Sec. 7.4 April 2, / 12
12 Summary 1 Any set of n linearly independent solutions of the system x = P(t)x constitutes a fundamental set of solution. 2 Under the conditions given in this section, such fundamental sets always exists. 3 Every solution of the system x = P(t)x can be represented as a linear combination of any fundamental set of solutions. MATH 351 (Differential Equations) Sec. 7.4 April 2, / 12
Chapter 4: Higher Order Linear Equations
Chapter 4: Higher Order Linear Equations MATH 351 California State University, Northridge April 7, 2014 MATH 351 (Differential Equations) Ch 4 April 7, 2014 1 / 11 Sec. 4.1: General Theory of nth Order
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 information10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections )
c Dr. Igor Zelenko, Fall 2017 1 10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections 7.2-7.4) 1. When each of the functions F 1, F 2,..., F n in right-hand side
More informationSecond Order Linear Equations
October 13, 2016 1 Second And Higher Order Linear Equations In first part of this chapter, we consider second order linear ordinary linear equations, i.e., a differential equation of the form L[y] = d
More informationMath 331 Homework Assignment Chapter 7 Page 1 of 9
Math Homework Assignment Chapter 7 Page of 9 Instructions: Please make sure to demonstrate every step in your calculations. Return your answers including this homework sheet back to the instructor as a
More information4. Higher Order Linear DEs
4. Higher Order Linear DEs Department of Mathematics & Statistics ASU Outline of Chapter 4 1 General Theory of nth Order Linear Equations 2 Homogeneous Equations with Constant Coecients 3 The Method of
More informationLinear Independence and the Wronskian
Linear Independence and the Wronskian MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 Operator Notation Let functions p(t) and q(t) be continuous functions
More informationMATH 308 Differential Equations
MATH 308 Differential Equations Summer, 2014, SET 5 JoungDong Kim Set 5: Section 3.1, 3.2 Chapter 3. Second Order Linear Equations. Section 3.1 Homogeneous Equations with Constant Coefficients. In this
More informationLecture 16. Theory of Second Order Linear Homogeneous ODEs
Math 245 - Mathematics of Physics and Engineering I Lecture 16. Theory of Second Order Linear Homogeneous ODEs February 17, 2012 Konstantin Zuev (USC) Math 245, Lecture 16 February 17, 2012 1 / 12 Agenda
More informationLecture 9. Systems of Two First Order Linear ODEs
Math 245 - Mathematics of Physics and Engineering I Lecture 9. Systems of Two First Order Linear ODEs January 30, 2012 Konstantin Zuev (USC) Math 245, Lecture 9 January 30, 2012 1 / 15 Agenda General Form
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 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 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 informationHigher Order Linear Equations Lecture 7
Higher Order Linear Equations Lecture 7 Dibyajyoti Deb 7.1. Outline of Lecture General Theory of nth Order Linear Equations. Homogeneous Equations with Constant Coefficients. 7.2. General Theory of nth
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 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 informationApplied Differential Equation. November 30, 2012
Applied Differential Equation November 3, Contents 5 System of First Order Linear Equations 5 Introduction and Review of matrices 5 Systems of Linear Algebraic Equations, Linear Independence, Eigenvalues,
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 informationSystems of Ordinary Differential Equations
Systems of Ordinary Differential Equations MATH 365 Ordinary Differential Equations J Robert Buchanan Department of Mathematics Fall 2018 Objectives Many physical problems involve a number of separate
More informationMath 54. Selected Solutions for Week 5
Math 54. Selected Solutions for Week 5 Section 4. (Page 94) 8. Consider the following two systems of equations: 5x + x 3x 3 = 5x + x 3x 3 = 9x + x + 5x 3 = 4x + x 6x 3 = 9 9x + x + 5x 3 = 5 4x + x 6x 3
More informationSept. 3, 2013 Math 3312 sec 003 Fall 2013
Sept. 3, 2013 Math 3312 sec 003 Fall 2013 Section 1.8: Intro to Linear Transformations Recall that the product Ax is a linear combination of the columns of A turns out to be a vector. If the columns of
More informationSection 1.4: Second-Order and Higher-Order Equations. Consider a second-order, linear, homogeneous equation with constant coefficients
Section 1.4: Second-Order and Higher-Order Equations Consider a second-order, linear, homogeneous equation with constant coefficients x t+2 + ax t+1 + bx t = 0. (1) To solve this difference equation, we
More informationOptimal Control. Quadratic Functions. Single variable quadratic function: Multi-variable quadratic function:
Optimal Control Control design based on pole-placement has non unique solutions Best locations for eigenvalues are sometimes difficult to determine Linear Quadratic LQ) Optimal control minimizes a quadratic
More informationODEs Cathal Ormond 1
ODEs Cathal Ormond 2 1. Separable ODEs Contents 2. First Order ODEs 3. Linear ODEs 4. 5. 6. Chapter 1 Separable ODEs 1.1 Definition: An ODE An Ordinary Differential Equation (an ODE) is an equation whose
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 information20D - Homework Assignment 5
Brian Bowers TA for Hui Sun MATH D Homework Assignment 5 November 8, 3 D - Homework Assignment 5 First, I present the list of all matrix row operations. We use combinations of these steps to row reduce
More informationMODULE 13. Topics: Linear systems
Topics: Linear systems MODULE 13 We shall consider linear operators and the associated linear differential equations. Specifically we shall have operators of the form i) Lu u A(t)u where A(t) is an n n
More informationVANDERBILT UNIVERSITY. MATH 2610 ORDINARY DIFFERENTIAL EQUATIONS Practice for test 1 solutions
VANDERBILT UNIVERSITY MATH 2610 ORDINARY DIFFERENTIAL EQUATIONS Practice for test 1 solutions The first test will cover all material discussed up to (including) section 4.5. Important: The solutions below
More informationDifferential equations
Differential equations Math 7 Spring Practice problems for April Exam Problem Use the method of elimination to find the x-component of the general solution of x y = 6x 9x + y = x 6y 9y Soln: The system
More information2nd-Order Linear Equations
4 2nd-Order Linear Equations 4.1 Linear Independence of Functions In linear algebra the notion of linear independence arises frequently in the context of vector spaces. If V is a vector space over the
More informationA Second Course in Elementary Differential Equations
A Second Course in Elementary Differential Equations Marcel B Finan Arkansas Tech University c All Rights Reserved August 3, 23 Contents 28 Calculus of Matrix-Valued Functions of a Real Variable 4 29 nth
More informationMath 211. Lecture #6. Linear Equations. September 9, 2002
1 Math 211 Lecture #6 Linear Equations September 9, 2002 2 Air Resistance 2 Air Resistance Acts in the direction opposite to the velocity. 2 Air Resistance Acts in the direction opposite to the velocity.
More informationPartial proof: y = ϕ 1 (t) is a solution to y + p(t)y = 0 implies. Thus y = cϕ 1 (t) is a solution to y + p(t)y = 0 since
Existence and Uniqueness for LINEAR DEs. Homogeneous: y (n) + p 1 (t)y (n 1) +...p n 1 (t)y + p n (t)y = 0 Non-homogeneous: g(t) 0 y (n) + p 1 (t)y (n 1) +...p n 1 (t)y + p n (t)y = g(t) 1st order LINEAR
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 information2.3. VECTOR SPACES 25
2.3. VECTOR SPACES 25 2.3 Vector Spaces MATH 294 FALL 982 PRELIM # 3a 2.3. Let C[, ] denote the space of continuous functions defined on the interval [,] (i.e. f(x) is a member of C[, ] if f(x) is continuous
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 informationAFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda 4. BASES AND DIMENSION
4. BASES AND DIMENSION Definition Let u 1,..., u n be n vectors in V. The vectors u 1,..., u n are linearly independent if the only linear combination of them equal to the zero vector has only zero scalars;
More informationPRELIMINARY THEORY LINEAR EQUATIONS
4.1 PRELIMINARY THEORY LINEAR EQUATIONS 117 4.1 PRELIMINARY THEORY LINEAR EQUATIONS REVIEW MATERIAL Reread the Remarks at the end of Section 1.1 Section 2.3 (especially page 57) INTRODUCTION In Chapter
More informationµ = e R p(t)dt where C is an arbitrary constant. In the presence of an initial value condition
MATH 3860 REVIEW FOR FINAL EXAM The final exam will be comprehensive. It will cover materials from the following sections: 1.1-1.3; 2.1-2.2;2.4-2.6;3.1-3.7; 4.1-4.3;6.1-6.6; 7.1; 7.4-7.6; 7.8. The following
More informationFirst and Second Order Differential Equations Lecture 4
First and Second Order Differential Equations Lecture 4 Dibyajyoti Deb 4.1. Outline of Lecture The Existence and the Uniqueness Theorem Homogeneous Equations with Constant Coefficients 4.2. The Existence
More informationMath 3313: Differential Equations Second-order ordinary differential equations
Math 3313: Differential Equations Second-order ordinary differential equations Thomas W. Carr Department of Mathematics Southern Methodist University Dallas, TX Outline Mass-spring & Newton s 2nd law Properties
More informationMath Ordinary Differential Equations
Math 411 - Ordinary Differential Equations Review Notes - 1 1 - Basic Theory A first order ordinary differential equation has the form x = f(t, x) (11) Here x = dx/dt Given an initial data x(t 0 ) = x
More informationMath 550 Notes. Chapter 2. Jesse Crawford. Department of Mathematics Tarleton State University. Fall 2010
Math 550 Notes Chapter 2 Jesse Crawford Department of Mathematics Tarleton State University Fall 2010 (Tarleton State University) Math 550 Chapter 2 Fall 2010 1 / 20 Linear algebra deals with finite dimensional
More informationSecond Order and Higher Order Equations Introduction
Second Order and Higher Order Equations Introduction Second order and higher order equations occur frequently in science and engineering (like pendulum problem etc.) and hence has its own importance. It
More information9 - Matrix Methods for Linear Systems
9 - Matrix Methods for Linear Systems 9.4 Linear Systems in Normal Form Homework: p. 523-526 # ü Introduction Consider a system of n linear differential equations given by x 1 x 2 x n = a 11 HtL x 1 HtL
More informationMath 308 Final Exam Practice Problems
Math 308 Final Exam Practice Problems This review should not be used as your sole source for preparation for the exam You should also re-work all examples given in lecture and all suggested homework problems
More informationAlgebraic Properties of Solutions of Linear Systems
Algebraic Properties of Solutions of Linear Systems In this chapter we will consider simultaneous first-order differential equations in several variables, that is, equations of the form f 1t,,,x n d f
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 informationRow Space, Column Space, and Nullspace
Row Space, Column Space, and Nullspace MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Every matrix has associated with it three vector spaces: row space
More informationSection 3.1 Second Order Linear Homogeneous DEs with Constant Coefficients
Section 3. Second Order Linear Homogeneous DEs with Constant Coefficients Key Terms/ Ideas: Initial Value Problems Homogeneous DEs with Constant Coefficients Characteristic equation Linear DEs of second
More informationPROBLEMS In each of Problems 1 through 12:
6.5 Impulse Functions 33 which is the formal solution of the given problem. It is also possible to write y in the form 0, t < 5, y = 5 e (t 5/ sin 5 (t 5, t 5. ( The graph of Eq. ( is shown in Figure 6.5.3.
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 informationReview of Linear Algebra
Review of Linear Algebra Definitions An m n (read "m by n") matrix, is a rectangular array of entries, where m is the number of rows and n the number of columns. 2 Definitions (Con t) A is square if m=
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 informationSecond Order Differential Equations Lecture 6
Second Order Differential Equations Lecture 6 Dibyajyoti Deb 6.1. Outline of Lecture Repeated Roots; Reduction of Order Nonhomogeneous Equations; Method of Undetermined Coefficients Variation of Parameters
More informationMath53: Ordinary Differential Equations Autumn 2004
Math53: Ordinary Differential Equations Autumn 2004 Unit 2 Summary Second- and Higher-Order Ordinary Differential Equations Extremely Important: Euler s formula Very Important: finding solutions to linear
More informationSection 9.2: Matrices. Definition: A matrix A consists of a rectangular array of numbers, or elements, arranged in m rows and n columns.
Section 9.2: Matrices Definition: A matrix A consists of a rectangular array of numbers, or elements, arranged in m rows and n columns. That is, a 11 a 12 a 1n a 21 a 22 a 2n A =...... a m1 a m2 a mn A
More informationSolutions to Homework 5 - Math 3410
Solutions to Homework 5 - Math 34 (Page 57: # 489) Determine whether the following vectors in R 4 are linearly dependent or independent: (a) (, 2, 3, ), (3, 7,, 2), (, 3, 7, 4) Solution From x(, 2, 3,
More informationV 1 V 2. r 3. r 6 r 4. Math 2250 Lab 12 Due Date : 4/25/2017 at 6:00pm
Math 50 Lab 1 Name: Due Date : 4/5/017 at 6:00pm 1. In the previous lab you considered the input-output model below with pure water flowing into the system, C 1 = C 5 =0. r 1, C 1 r 5, C 5 r r V 1 V r
More information(i) [7 points] Compute the determinant of the following matrix using cofactor expansion.
Question (i) 7 points] Compute the determinant of the following matrix using cofactor expansion 2 4 2 4 2 Solution: Expand down the second column, since it has the most zeros We get 2 4 determinant = +det
More informationMTH 5102 Linear Algebra Practice Final Exam April 26, 2016
Name (Last name, First name): MTH 5 Linear Algebra Practice Final Exam April 6, 6 Exam Instructions: You have hours to complete the exam. There are a total of 9 problems. You must show your work and write
More informationc 1 = y 0, c 2 = 1 2 y 1. Therefore the solution to the general initial-value problem is y(t) = y 0 cos(2t)+y sin(2t).
Solutions to Second In-Class Exam Math 246, Professor David Levermore Tuesday, 29 October 2 ( [4] Give the interval of definition for the solution of the initial-value problem u t u + cos(5t 6+t u = et
More informationLecture Notes for Math 524
Lecture Notes for Math 524 Dr Michael Y Li October 19, 2009 These notes are based on the lecture notes of Professor James S Muldowney, the books of Hale, Copple, Coddington and Levinson, and Perko They
More informationMATH Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product.
MATH 311-504 Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product. Determinant is a scalar assigned to each square matrix. Notation. The determinant of a matrix A = (a ij
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 informationMATH 174A: PROBLEM SET 3. Suggested Solution. det(a + tb) = (det A) det(i + ta 1 B)
MATH 174A: PROBLEM SET 3 Suggested Solution Problem 1. (Cf. Taylor I.1.3.) Let M n n (C) denote the set of n n complex matrices. Suppose A M n n (C) is invertible. Using show that det(a + tb) = (det A)
More informationModal Decomposition and the Time-Domain Response of Linear Systems 1
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING.151 Advanced System Dynamics and Control Modal Decomposition and the Time-Domain Response of Linear Systems 1 In a previous handout
More informationSolution to Homework 1
Solution to Homework Sec 2 (a) Yes It is condition (VS 3) (b) No If x, y are both zero vectors Then by condition (VS 3) x = x + y = y (c) No Let e be the zero vector We have e = 2e (d) No It will be false
More informationChapter 1: Systems of Linear Equations
Chapter : Systems of Linear Equations February, 9 Systems of linear equations Linear systems Lecture A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where
More informationShort Review of Basic Mathematics
Short Review of Basic Mathematics Tomas Co January 1, 1 c 8 Tomas Co, all rights reserve Contents 1 Review of Functions 5 11 Mathematical Ientities 5 1 Drills 7 1 Answers to Drills 8 Review of ODE Solutions
More informationMathematics 22: Lecture 12
Mathematics 22: Lecture 12 Second-order Linear Equations Dan Sloughter Furman University January 28, 2008 Dan Sloughter (Furman University) Mathematics 22: Lecture 12 January 28, 2008 1 / 14 Definition
More informationSection 9.2: Matrices.. a m1 a m2 a mn
Section 9.2: Matrices Definition: A matrix is a rectangular array of numbers: a 11 a 12 a 1n a 21 a 22 a 2n A =...... a m1 a m2 a mn In general, a ij denotes the (i, j) entry of A. That is, the entry in
More informationUnderstand the existence and uniqueness theorems and what they tell you about solutions to initial value problems.
Review Outline To review for the final, look over the following outline and look at problems from the book and on the old exam s and exam reviews to find problems about each of the following topics.. Basics
More information2.3 Terminology for Systems of Linear Equations
page 133 e 2t sin 2t 44 A(t) = t 2 5 te t, a = 0, b = 1 sec 2 t 3t sin t 45 The matrix function A(t) in Problem 39, with a = 0 and b = 1 Integration of matrix functions given in the text was done with
More informationSection 2.8: The Existence and Uniqueness Theorem
Section 2.8: The Existence and Uniqueness Theorem MATH 351 California State University, Northridge March 10, 2014 MATH 351 (Differetial Equations) Sec. 2.8 March 10, 2014 1 / 26 Theorem 2.4.2 in Section
More informationWeek 9-10: Recurrence Relations and Generating Functions
Week 9-10: Recurrence Relations and Generating Functions April 3, 2017 1 Some number sequences An infinite sequence (or just a sequence for short is an ordered array a 0, a 1, a 2,..., a n,... of countably
More informationMath 240 Calculus III
DE Higher Order Calculus III Summer 2015, Session II Tuesday, July 28, 2015 Agenda DE 1. of order n An example 2. constant-coefficient linear Introduction DE We now turn our attention to solving linear
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 informationChoose three of: Choose three of: Choose three of:
MATH Final Exam (Version ) Solutions July 8, 8 S. F. Ellermeyer Name Instructions. Remember to include all important details of your work. You will not get full credit (or perhaps even any partial credit)
More informationMATH 12 CLASS 5 NOTES, SEP
MATH 12 CLASS 5 NOTES, SEP 30 2011 Contents 1. Vector-valued functions 1 2. Differentiating and integrating vector-valued functions 3 3. Velocity and Acceleration 4 Over the past two weeks we have developed
More informationMath 4B Notes. Written by Victoria Kala SH 6432u Office Hours: T 12:45 1:45pm Last updated 7/24/2016
Math 4B Notes Written by Victoria Kala vtkala@math.ucsb.edu SH 6432u Office Hours: T 2:45 :45pm Last updated 7/24/206 Classification of Differential Equations The order of a differential equation is the
More informationMATH 4B Differential Equations, Fall 2016 Final Exam Study Guide
MATH 4B Differential Equations, Fall 2016 Final Exam Study Guide GENERAL INFORMATION AND FINAL EXAM RULES The exam will have a duration of 3 hours. No extra time will be given. Failing to submit your solutions
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 information144 Chapter 3. Second Order Linear Equations
144 Chapter 3. Second Order Linear Equations PROBLEMS In each of Problems 1 through 8 find the general solution of the given differential equation. 1. y + 2y 3y = 0 2. y + 3y + 2y = 0 3. 6y y y = 0 4.
More informationSecond Order Linear Equations
Second Order Linear Equations Linear Equations The most general linear ordinary differential equation of order two has the form, a t y t b t y t c t y t f t. 1 We call this a linear equation because the
More informationEIGENVALUES AND EIGENVECTORS 3
EIGENVALUES AND EIGENVECTORS 3 1. Motivation 1.1. Diagonal matrices. Perhaps the simplest type of linear transformations are those whose matrix is diagonal (in some basis). Consider for example the matrices
More informationLec 2: Mathematical Economics
Lec 2: Mathematical Economics to Spectral Theory Sugata Bag Delhi School of Economics 24th August 2012 [SB] (Delhi School of Economics) Introductory Math Econ 24th August 2012 1 / 17 Definition: Eigen
More informationMIDTERM REVIEW AND SAMPLE EXAM. Contents
MIDTERM REVIEW AND SAMPLE EXAM Abstract These notes outline the material for the upcoming exam Note that the review is divided into the two main topics we have covered thus far, namely, ordinary differential
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 informationLinear Algebra MATH20F Midterm 1
University of California San Diego NAME TA: Linear Algebra Wednesday, October st, 9 :am - :5am No aids are allowed Be sure to write all row operations used Remember that you can often check your answers
More informationSecond-Order Linear ODEs
Second-Order Linear ODEs A second order ODE is called linear if it can be written as y + p(t)y + q(t)y = r(t). (0.1) It is called homogeneous if r(t) = 0, and nonhomogeneous otherwise. We shall assume
More informationAdvanced Mathematics for Economics, course Juan Pablo Rincón Zapatero
Advanced Mathematics for Economics, course 2013-2014 Juan Pablo Rincón Zapatero Contents 1. Review of Matrices and Determinants 2 1.1. Square matrices 2 1.2. Determinants 3 2. Diagonalization of matrices
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 informationHIGHER-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS. David Levermore Department of Mathematics University of Maryland.
HIGHER-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS David Levermore Department of Mathematics University of Maryland 28 March 2008 The following is a review of some of the material that we covered on higher-order
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 informationDepartment of Mathematics IIT Guwahati
Stability of Linear Systems in R 2 Department of Mathematics IIT Guwahati A system of first order differential equations is called autonomous if the system can be written in the form dx 1 dt = g 1(x 1,
More informationMATH 2250 Final Exam Solutions
MATH 225 Final Exam Solutions Tuesday, April 29, 28, 6: 8:PM Write your name and ID number at the top of this page. Show all your work. You may refer to one double-sided sheet of notes during the exam
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 informationChapter 7. Linear Algebra: Matrices, Vectors,
Chapter 7. Linear Algebra: Matrices, Vectors, Determinants. Linear Systems Linear algebra includes the theory and application of linear systems of equations, linear transformations, and eigenvalue problems.
More informationExercises Chapter II.
Page 64 Exercises Chapter II. 5. Let A = (1, 2) and B = ( 2, 6). Sketch vectors of the form X = c 1 A + c 2 B for various values of c 1 and c 2. Which vectors in R 2 can be written in this manner? B y
More information