Section 1.4: Second-Order and Higher-Order Equations. Consider a second-order, linear, homogeneous equation with constant coefficients
|
|
- Lorraine York
- 6 years ago
- Views:
Transcription
1 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 consider solutions of the form x t = λ t, where λ 0. Definition: The equation λ 2 +aλ+b = 0 is called the characteristic equation and λ 2 +aλ+b is called the characteristic polynomial of the difference equation (1). The two solutions of the characteristic equation are called the eigenvalues. The form of the general solution depends on the eigenvalues and can be separated into three cases. Case 1: Real Distinct Eigenvalues If the eigenvalues are real and distinct (λ 1 λ 2 ), then the general solution is x t = c 1 λ t 1 + c 2 λ t 2. Case 2: Real Repeated Eigenvalues If the eigenvalues are real and repeated (λ 1 = λ 2 ), then the general solution is x t = c 1 λ t 1 + c 2 tλ t 2. Case 3: Complex Conjugate Eigenvalues If the eigenvalues are complex conjugates (λ = a ± ib), then the general solution is where r = a 2 + b 2 and φ = arctan(b/a). x t = c 1 r t cos(φt) + c 2 r t sin(φt), In each case, the general solution is an arbitrary linear combination of two linearly-independent solutions x 1 (t) and x 2 (t). This is known as the superposition principle. 1
2 A check for linear independence involves a quantity called the Casoratian, which is very similar to the Wronskian in differential equations. Definition: The Casoratian of x 1 (t) and x 2 (t) is C(t) = C(x 1 (t), x 2 (t)) = x 1 (t) x 2 (t) x 1 (t + 1) x 2 (t + 1) = x 1(t)x 2 (t + 1) x 2 (t)x 1 (t + 1). The solutions x 1 (t) and x 2 (t) are linearly independent if C(t) 0 for some t = 0, 1, 2,.... In this case, x 1 (t) and x 2 (t) are said to form a fundamental set of solutions. Example: Find the Casoratian of each set of functions. (a) x 1 (t) = λ t 1, x 2 (t) = λ t 2 (b) x 1 (t) = λ t, x 2 (t) = tλ t 2
3 Example: Find the general solution of each difference equation. (a) x t+2 + 3x t+1 + 2x t = 0 (b) x t+2 10x t x t = 0 (c) x t+2 + 4x t+1 + 8x t = 0 3
4 Example: Solve the initial value problem x t+2 + 4x t+1 + 3x t = 0, x 0 = 0, x 1 = 1. Example: Solve the initial value problem x t+2 + 4x t+1 + 4x t = 0, x 0 = 2, x 1 = 4. 4
5 Higher-Order Linear Equations Consider a kth-order, linear, homogeneous difference equation with constant coefficients x t+k + a 1 x t+k a k x t = 0. (2) Let x t = λ t, where λ 0. The characteristic equation for this difference equation is λ k + a 1 λ k a k = 0. The characteristic equation has k eigenvalues λ 1,..., λ k. Case 1: Real Distinct Eigenvalues If the eigenvalues are real and distinct, then the general solution is x t = c 1 λ t c k λ t k. Case 2: Real Repeated Eigenvalues If there is a real eigenvalue of multiplicity m, then m linearly independent solutions can be formed as λ t 1, tλ t 1,..., t m 1 λ t 1. Case 3: Complex Conjugate Eigenvalues If there are complex conjugate eigenvalues λ 1,2 = r (cos φ ± i sin φ) of multiplicity m, then there are 2m linearly independent solutions r t cos(tφ), r t sin(tφ), tr t cos(tφ), tr t sin(tφ),..., t m 1 r t cos(tφ), t m 1 r t sin(tφ). In this case the Casoratian is the determinant of a k k matrix and the solutions can be shown to be linearly independent. Example: Find the general solution of x t+3 + x t+2 + x t+1 + x t = 0. 5
6 Nonhomogeneous Linear Equations Consider a kth-order, nonhomogeneous, linear difference equation where b t is a nonzero function of t = 0, 1, 2,.... x t+k + a 1 x t+k a k x t = b t, (3) The general solution of the nonhomogeneous equation is given by x(t) = x h (t) + x p (t), where x h (t) is a solution of the corresponding homogeneous equation and x p (t) is a particular solution of the nonhomogeneous equation. To find a particular solution, we use the Method of Undetermined Coefficients. That is, we assume the particular solution has the same form as the forcing term b t. Example: Find the general solution of the nonhomogeneous equation x t+2 5x t+1 + 6x t = t
7 Example: Find the general solution of the nonhomogeneous equation x t+2 + x t+1 12x t = t2 t. 7
8 Limiting Behavior of Solutions The long-term behavior of solutions of linear difference equations is determined by the eigenvalues of the characteristic equation. The magnitude of the eigenvalues determine whether solutions are bounded or unbounded. The types of eigenvalues (real or complex) determine whether solutions oscillate or whether solutions converge or diverge monotonically. Of particular interest is whether the magnitude of all eigenvalues is less than one. Recall that the magnitude of a real eigenvalue λ = a is its absolute value λ = a. The magnitude of a complex eigenvlaue λ = a + bi is λ = a + bi = a 2 + b 2. Definition: Suppose that the k eigenvalues of a characteristic equation are λ 1,..., λ k. An eigenvalue λ i such that λ i λ j for all j i is called a dominant eigenvalue. If the inequality is strict, λ i > λ j for all j i, then λ i is called a strictly dominant eigenvalue. In particular, if there exists a dominant eigenvalue λ 1 such that λ 1 < 1, then solutions to the difference equation converge to zero. Example: Consider the difference equation 4x t+2 + x t = 0. (a) Find the eigenvalues of this difference equation. Identify the dominant eigenvalue. Is it strictly dominant? (b) Determine the limiting behavior of x t without solving the equation. (c) Find the general solution of the difference equation. limiting behavior you expected? Does the solution exhibit the 8
Higher 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 informationTheory of Higher-Order Linear Differential Equations
Chapter 6 Theory of Higher-Order Linear Differential Equations 6.1 Basic Theory A linear differential equation of order n has the form a n (x)y (n) (x) + a n 1 (x)y (n 1) (x) + + a 0 (x)y(x) = b(x), (6.1.1)
More informationMath 2142 Homework 5 Part 1 Solutions
Math 2142 Homework 5 Part 1 Solutions Problem 1. For the following homogeneous second order differential equations, give the general solution and the particular solution satisfying the given initial conditions.
More informationLINEAR EQUATIONS OF HIGHER ORDER. EXAMPLES. General framework
Differential Equations Grinshpan LINEAR EQUATIONS OF HIGHER ORDER. EXAMPLES. We consider linear ODE of order n: General framework (1) x (n) (t) + P n 1 (t)x (n 1) (t) + + P 1 (t)x (t) + P 0 (t)x(t) = 0
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 informationHomework 3 Solutions Math 309, Fall 2015
Homework 3 Solutions Math 39, Fall 25 782 One easily checks that the only eigenvalue of the coefficient matrix is λ To find the associated eigenvector, we have 4 2 v v 8 4 (up to scalar multiplication)
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 informationSection 9.8 Higher Order Linear Equations
Section 9.8 Higher Order Linear Equations Key Terms: Higher order linear equations Equivalent linear systems for higher order equations Companion matrix Characteristic polynomial and equation A linear
More informationChapter 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 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 informationLinear differential equations with constant coefficients Method of undetermined coefficients
Linear differential equations with constant coefficients Method of undetermined coefficients e u+vi = e u (cos vx + i sin vx), u, v R, i 2 = -1 Quasi-polynomial: Q α+βi,k (x) = e αx [cos βx ( f 0 + f 1
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 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 informationThe Corrected Trial Solution in the Method of Undetermined Coefficients
Definition of Related Atoms The Basic Trial Solution Method Symbols Superposition Annihilator Polynomial for f(x) Annihilator Equation for f(x) The Corrected Trial Solution in the Method of Undetermined
More informationNonhomogeneous Linear Differential Equations with Constant Coefficients - (3.4) Method of Undetermined Coefficients
Nonhomogeneous Linear Differential Equations with Constant Coefficients - (3.4) Method of Undetermined Coefficients Consider an nth-order nonhomogeneous linear differential equation with constant coefficients:
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 informationHow to Solve Linear Differential Equations
How to Solve Linear Differential Equations Definition: Euler Base Atom, Euler Solution Atom Independence of Atoms Construction of the General Solution from a List of Distinct Atoms Euler s Theorems Euler
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 informationNotation. 0,1,2,, 1 with addition and multiplication modulo
Notation Q,, The set of all natural numbers 1,2,3, The set of all integers The set of all rational numbers The set of all real numbers The group of permutations of distinct symbols 0,1,2,,1 with addition
More informationMa 227 Review for Systems of DEs
Ma 7 Review for Systems of DEs Matrices Basic Properties Addition and subtraction: Let A a ij mn and B b ij mn.then A B a ij b ij mn 3 A 6 B 6 4 7 6 A B 6 4 3 7 6 6 7 3 Scaler Multiplication: Let k be
More informationMATH 24 EXAM 3 SOLUTIONS
MATH 4 EXAM 3 S Consider the equation y + ω y = cosω t (a) Find the general solution of the homogeneous equation (b) Find the particular solution of the non-homogeneous equation using the method of Undetermined
More informationAPPM 2360 Section Exam 3 Wednesday November 19, 7:00pm 8:30pm, 2014
APPM 2360 Section Exam 3 Wednesday November 9, 7:00pm 8:30pm, 204 ON THE FRONT OF YOUR BLUEBOOK write: () your name, (2) your student ID number, (3) lecture section, (4) your instructor s name, and (5)
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 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 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 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 informationMA 266 Review Topics - Exam # 2 (updated)
MA 66 Reiew Topics - Exam # updated Spring First Order Differential Equations Separable, st Order Linear, Homogeneous, Exact Second Order Linear Homogeneous with Equations Constant Coefficients The differential
More information6 - Theory of Higher-Order Linear Differential Equations
6 - Theory of Higher-Order Linear Differential Equations 6.1 Basic Theory of Linear Differential Equations Homework: p. 325-326 #1, 7, 15, 19 ü Introduction A linear differential equation of order n is
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 informationMA22S3 Summary Sheet: Ordinary Differential Equations
MA22S3 Summary Sheet: Ordinary Differential Equations December 14, 2017 Kreyszig s textbook is a suitable guide for this part of the module. Contents 1 Terminology 1 2 First order separable 2 2.1 Separable
More informationStep 1. Step 2. Step 4. The corrected trial solution y with evaluated coefficients d 1, d 2,..., d k becomes the particular solution y p.
Definition Atoms A and B are related if and only if their successive derivatives share a common atom. Then x 3 is related to x and x 101, while x is unrelated to e x, xe x and x sin x. Atoms x sin x and
More informationSection 5.5. Complex Eigenvalues
Section 5.5 Complex Eigenvalues Motivation: Describe rotations Among transformations, rotations are very simple to describe geometrically. Where are the eigenvectors? A no nonzero vector x is collinear
More informationVector Spaces and Subspaces
Vector Spaces and Subspaces Vector Space V Subspaces S of Vector Space V The Subspace Criterion Subspaces are Working Sets The Kernel Theorem Not a Subspace Theorem Independence and Dependence in Abstract
More informationMath 54. Selected Solutions for Week 10
Math 54. Selected Solutions for Week 10 Section 4.1 (Page 399) 9. Find a synchronous solution of the form A cos Ωt+B sin Ωt to the given forced oscillator equation using the method of Example 4 to solve
More informationsystems of linear di erential If the homogeneous linear di erential system is diagonalizable,
G. NAGY ODE October, 8.. Homogeneous Linear Differential Systems Section Objective(s): Linear Di erential Systems. Diagonalizable Systems. Real Distinct Eigenvalues. Complex Eigenvalues. Repeated Eigenvalues.
More informationChapter Homogeneous System with Constant Coeffici
Chapter 7 7.5 Homogeneous System with Constant Coefficients Homogeneous System with constant coefficients We consider homogeneous linear systems: x = Ax A is an n n matrix with constant entries () As in
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 informationBasic Theory of Linear Differential Equations
Basic Theory of Linear Differential Equations Picard-Lindelöf Existence-Uniqueness Vector nth Order Theorem Second Order Linear Theorem Higher Order Linear Theorem Homogeneous Structure Recipe for Constant-Coefficient
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 informationLinear Differential Equations. Problems
Chapter 1 Linear Differential Equations. Problems 1.1 Introduction 1.1.1 Show that the function ϕ : R R, given by the expression ϕ(t) = 2e 3t for all t R, is a solution of the Initial Value Problem x =
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 information1+t 2 (l) y = 2xy 3 (m) x = 2tx + 1 (n) x = 2tx + t (o) y = 1 + y (p) y = ty (q) y =
DIFFERENTIAL EQUATIONS. Solved exercises.. Find the set of all solutions of the following first order differential equations: (a) x = t (b) y = xy (c) x = x (d) x = (e) x = t (f) x = x t (g) x = x log
More informationDifferential Equations Practice: 2nd Order Linear: Nonhomogeneous Equations: Undetermined Coefficients Page 1
Differential Equations Practice: 2nd Order Linear: Nonhomogeneous Equations: Undetermined Coefficients Page 1 Questions Example (3.5.3) Find a general solution of the differential equation y 2y 3y = 3te
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 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 informationSec. 7.4: Basic Theory of Systems of First Order Linear Equations
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, 214 1 / 12 System of
More informationCh 10.1: Two Point Boundary Value Problems
Ch 10.1: Two Point Boundary Value Problems In many important physical problems there are two or more independent variables, so the corresponding mathematical models involve partial differential equations.
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 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 informationANSWERS Final Exam Math 250b, Section 2 (Professor J. M. Cushing), 15 May 2008 PART 1
ANSWERS Final Exam Math 50b, Section (Professor J. M. Cushing), 5 May 008 PART. (0 points) A bacterial population x grows exponentially according to the equation x 0 = rx, where r>0is the per unit rate
More informationSection 3.4. Second Order Nonhomogeneous. The corresponding homogeneous equation
Section 3.4. Second Order Nonhomogeneous Equations y + p(x)y + q(x)y = f(x) (N) The corresponding homogeneous equation y + p(x)y + q(x)y = 0 (H) is called the reduced equation of (N). 1 General Results
More informationSection 3.4. Second Order Nonhomogeneous. The corresponding homogeneous equation. is called the reduced equation of (N).
Section 3.4. Second Order Nonhomogeneous Equations y + p(x)y + q(x)y = f(x) (N) The corresponding homogeneous equation y + p(x)y + q(x)y = 0 (H) is called the reduced equation of (N). 1 General Results
More informationSECOND ORDER ODE S. 1. A second order differential equation is an equation of the form. F (x, y, y, y ) = 0.
SECOND ORDER ODE S 1. A second der differential equation is an equation of the fm F (x, y, y, y ) = 0. A solution of the differential equation is a function y = y(x) that satisfies the equation. A differential
More information3.5 Undetermined Coefficients
3.5. UNDETERMINED COEFFICIENTS 153 11. t 2 y + ty + 4y = 0, y(1) = 3, y (1) = 4 12. t 2 y 4ty + 6y = 0, y(0) = 1, y (0) = 1 3.5 Undetermined Coefficients In this section and the next we consider the nonhomogeneous
More informationMath Assignment 5
Math 2280 - Assignment 5 Dylan Zwick Fall 2013 Section 3.4-1, 5, 18, 21 Section 3.5-1, 11, 23, 28, 35, 47, 56 Section 3.6-1, 2, 9, 17, 24 1 Section 3.4 - Mechanical Vibrations 3.4.1 - Determine the period
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 informationSection 9.3 Phase Plane Portraits (for Planar Systems)
Section 9.3 Phase Plane Portraits (for Planar Systems) Key Terms: Equilibrium point of planer system yꞌ = Ay o Equilibrium solution Exponential solutions o Half-line solutions Unstable solution Stable
More information1. TRUE or FALSE. 2. Find the complete solution set to the system:
TRUE or FALSE (a A homogenous system with more variables than equations has a nonzero solution True (The number of pivots is going to be less than the number of columns and therefore there is a free variable
More informationAdditional Homework Problems
Math 216 2016-2017 Fall Additional Homework Problems 1 In parts (a) and (b) assume that the given system is consistent For each system determine all possibilities for the numbers r and n r where r is the
More informationChapter 2. Linear Differential Equation of Second (or Higher) Order
Chapter 2. Linear Differential Equation of Second (or Higher) Order Contents: Homogeneous Linear Equations of Second Order (Section 2.1) Second-Order Homogeneous Linear Equation with Constant Coefficients
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 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 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 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 informationMA 1: PROBLEM SET NO. 7 SOLUTIONS
MA 1: PROBLEM SET NO. 7 SOLUTIONS 1. (1) Obtain the number r = 15 3 as an approimation to the nonzero root of the equation 2 = sin() by using the cubic Taylor polynomial approimation to sin(). Proof. The
More informationExamples: Solving nth Order Equations
Atoms L. Euler s Theorem The Atom List First Order. Solve 2y + 5y = 0. Examples: Solving nth Order Equations Second Order. Solve y + 2y + y = 0, y + 3y + 2y = 0 and y + 2y + 5y = 0. Third Order. Solve
More informationOrdinary Differential Equations
Ordinary Differential Equations (MA102 Mathematics II) Shyamashree Upadhyay IIT Guwahati Shyamashree Upadhyay ( IIT Guwahati ) Ordinary Differential Equations 1 / 15 Method of Undetermined Coefficients
More informationCopyright (c) 2006 Warren Weckesser
2.2. PLANAR LINEAR SYSTEMS 3 2.2. Planar Linear Systems We consider the linear system of two first order differential equations or equivalently, = ax + by (2.7) dy = cx + dy [ d x x = A x, where x =, and
More informationMA 262, Fall 2017, Final Version 01(Green)
INSTRUCTIONS MA 262, Fall 2017, Final Version 01(Green) (1) Switch off your phone upon entering the exam room. (2) Do not open the exam booklet until you are instructed to do so. (3) Before you open the
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 5. Higher Order Linear ODE'S
A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 2 A COLLECTION OF HANDOUTS ON SCALAR LINEAR ORDINARY
More informationSection 5.2 Solving Recurrence Relations
Section 5.2 Solving Recurrence Relations If a g(n) = f (a g(0),a g(1),..., a g(n 1) ) find a closed form or an expression for a g(n). Recall: nth degree polynomials have n roots: a n x n + a n 1 x n 1
More informationLinear Algebra- Final Exam Review
Linear Algebra- Final Exam Review. Let A be invertible. Show that, if v, v, v 3 are linearly independent vectors, so are Av, Av, Av 3. NOTE: It should be clear from your answer that you know the definition.
More informationP3.C8.COMPLEX NUMBERS
Recall: Within the real number system, we can solve equation of the form and b 2 4ac 0. ax 2 + bx + c =0, where a, b, c R What is R? They are real numbers on the number line e.g: 2, 4, π, 3.167, 2 3 Therefore,
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 information3. Identify and find the general solution of each of the following first order differential equations.
Final Exam MATH 33, Sample Questions. Fall 6. y = Cx 3 3 is the general solution of a differential equation. Find the equation. Answer: y = 3y + 9 xy. y = C x + C is the general solution of a differential
More informationTest #2 Math 2250 Summer 2003
Test #2 Math 225 Summer 23 Name: Score: There are six problems on the front and back of the pages. Each subpart is worth 5 points. Show all of your work where appropriate for full credit. ) Show the following
More informationMidterm for Introduction to Numerical Analysis I, AMSC/CMSC 466, on 10/29/2015
Midterm for Introduction to Numerical Analysis I, AMSC/CMSC 466, on 10/29/2015 The test lasts 1 hour and 15 minutes. No documents are allowed. The use of a calculator, cell phone or other equivalent electronic
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 informationAPPM 2360: Midterm 3 July 12, 2013.
APPM 2360: Midterm 3 July 12, 2013. ON THE FRONT OF YOUR BLUEBOOK write: (1) your name, (2) your instructor s name, (3) your recitation section number and (4) a grading table. Text books, class notes,
More informationWork sheet / Things to know. Chapter 3
MATH 251 Work sheet / Things to know 1. Second order linear differential equation Standard form: Chapter 3 What makes it homogeneous? We will, for the most part, work with equations with constant coefficients
More informationLecture Notes for Math 251: ODE and PDE. Lecture 30: 10.1 Two-Point Boundary Value Problems
Lecture Notes for Math 251: ODE and PDE. Lecture 30: 10.1 Two-Point Boundary Value Problems Shawn D. Ryan Spring 2012 Last Time: We finished Chapter 9: Nonlinear Differential Equations and Stability. Now
More informationSystems of Linear Differential Equations Chapter 7
Systems of Linear Differential Equations Chapter 7 Doreen De Leon Department of Mathematics, California State University, Fresno June 22, 25 Motivating Examples: Applications of Systems of First Order
More informationContents. 6 Systems of First-Order Linear Dierential Equations. 6.1 General Theory of (First-Order) Linear Systems
Dierential Equations (part 3): Systems of First-Order Dierential Equations (by Evan Dummit, 26, v 2) Contents 6 Systems of First-Order Linear Dierential Equations 6 General Theory of (First-Order) Linear
More informationHigher Order Linear ODEs
im03.qxd 9/21/05 11:04 AM Page 59 CHAPTER 3 Higher Order Linear ODEs This chapter is new. Its material is a rearranged and somewhat extended version of material previously contained in some of the sections
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 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 informationConvex Functions. Daniel P. Palomar. Hong Kong University of Science and Technology (HKUST)
Convex Functions Daniel P. Palomar Hong Kong University of Science and Technology (HKUST) ELEC5470 - Convex Optimization Fall 2017-18, HKUST, Hong Kong Outline of Lecture Definition convex function Examples
More informationThe Method of Undetermined Coefficients.
The Method of Undetermined Coefficients. James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University May 24, 2017 Outline 1 Annihilators 2 Finding The
More informationHomework #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 informationWork sheet / Things to know. Chapter 3
MATH 251 Work sheet / Things to know 1. Second order linear differential equation Standard form: Chapter 3 What makes it homogeneous? We will, for the most part, work with equations with constant coefficients
More information1.1 Limits and Continuity. Precise definition of a limit and limit laws. Squeeze Theorem. Intermediate Value Theorem. Extreme Value Theorem.
STATE EXAM MATHEMATICS Variant A ANSWERS AND SOLUTIONS 1 1.1 Limits and Continuity. Precise definition of a limit and limit laws. Squeeze Theorem. Intermediate Value Theorem. Extreme Value Theorem. Definition
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 informationAPPM 2360: Midterm exam 3 April 19, 2017
APPM 36: Midterm exam 3 April 19, 17 On the front of your Bluebook write: (1) your name, () your instructor s name, (3) your lecture section number and (4) a grading table. Text books, class notes, cell
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 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 informationJune 2011 PURDUE UNIVERSITY Study Guide for the Credit Exam in (MA 262) Linear Algebra and Differential Equations
June 20 PURDUE UNIVERSITY Study Guide for the Credit Exam in (MA 262) Linear Algebra and Differential Equations The topics covered in this exam can be found in An introduction to differential equations
More informationHigher Order Linear ODEs
c03.qxd 6/18/11 2:57 PM Page 57 CHAPTER 3 Higher Order Linear ODEs Chapters 1 and 2 demonstrate and illustrate that first- and second-order ODEs are, by far, the most important ones in usual engineering
More informationLecture 11. Andrei Antonenko. February 26, Last time we studied bases of vector spaces. Today we re going to give some examples of bases.
Lecture 11 Andrei Antonenko February 6, 003 1 Examples of bases Last time we studied bases of vector spaces. Today we re going to give some examples of bases. Example 1.1. Consider the vector space P the
More information