Size: px
Start display at page:

Download ""

Transcription

1 Differential Equations Grinshpan Two-Dimensional Homogeneous Linear Systems with Constant Coefficients. Purely Imaginary Eigenvalues. Recall the equation mẍ+k = of a simple harmonic oscillator with frequency ω = Let y = ẋ denote the velocity. Then ẏ = ẍ = k and we obtain a system of two m first-order linear equations {ẋ = y ) ẏ = k m. This system is coupled. Our goal is to understand the motion of t),yt)) in the y-plane phase plane). At each point,, the system ) prescribes a velocity vector, ẋ ) y = ẏ ω 2. We may start by plotting the velocity vector field: k. m The trajectories appear to be closed this is a consequence of periodicity of motion), symmetric about the origin, and oriented clockwise. There are no straight-line solutions. The constant vector = ) clearly satisfies the system. This is the only fied/stationary point in the phase plane, it corresponds to the static equilibrium of the oscillator. The law of conservation of energy allows us to identify the trajectories. Indeed, the potential energy U = 2 k2 and the kinetic energy K = 2 mẋ2 of the spring-mass system must add up to a nonnegative constant: 2 k2 + 2 my2 = C. Hence those closed trajectories are concentric ellipses.

2 Actually, the equation ω 2 ẋ+yẏ = is a direct consequence of ). Integrating it with respect to t, we arrive at the same conclusion: ω 2 2 +y 2 = C. For every C >, this is an ellipse centered at the origin with semiais ratio : ω How to describe a trajectory as a function of t? Since ω, stays on a circle, we may write = Ccosαt)), y = Cωsinαt)), where αt) is the polar angle. Using the equation ẋ = y, we conclude that α = ω and so α = ωt+δ. It follows that the general solution of ) can be written in the form cosωt δ) = C, ωsinωt δ) where C and δ phase shift) are arbitrary parameters. A variant of the preceding parameterization may also be of use. Observe that cosωt) sinωt) 2) P = and Q = ω sinωt) ωcosωt) are solutions of ). They initiate respectively at P) = and Q) = ) and therefore generate all solutions of the system, 3) P +c 2 Q. ω), Indeed, any given initial condition,y ) is satisfied by taking P + y ω P and Q form a fundamental pair of solutions. 2 Q. One says that

3 The matri of the system, Comple-variable method A = ω 2, has trace tr A = and determinant deta = ω 2. The characteristic polynomial pr) = r 2 +ω 2 of A has purely imaginary roots, r = ±iω, the eigenvalues of A. The eigenvectors of A have comple components: ω 2 = iω, iω iω ω 2 = iω iω. iω The eigenvectors determine a pair of independent straight-line in the comple sense!) solutions U = e iωt, V = e iω iωt. iω Separating real and imaginary parts we have cosωt) sinωt) cosωt) sinωt) U = +i, V = i. ω sinωt) ωcosωt) ω sinωt) ωcosωt) Thus U = P +iq and V = P iq are comple linear combinations of real solutions P and Q 2). The general solution with real components is given, as in 3), by P +c 2 Q, where c,c 2 are arbitrary real coefficients. Systems with purely imaginary eigenvalues The above ideas apply to any system {ẋ = a+by ẏ +dy whose matri has purely imaginary eigenvalues, r = ±iω. This is the case of zero trace a+d = and positive determinant ad bc >. We have ẋ ) = ẏ a b, a c a) y 2 bc = ω 2. The nonconstant trajectories are given by concentric ellipses a 2 +ω 2 ) 2 +2aby +b 2 y 2 = C. The frequency of rotation is given by ω. The fied point,) is referred to as center. It is neutral neither attracting nor repelling) in the sense of stability. 3

4 EXAMPLE. Consider the system {ẋ = 2y ẏ = 5 y. The underlying matri has zero trace and positive determinant, its eigenvalues are λ = ±3i. Hence we are dealing with a center. Let s find the solution in three ways. Approach. Reduce the system to a single second order equation. Solving the first equation for y, y = ẋ+, and substituting the result into the second equation deduce that ẍ+ ẋ = 5+ ẋ or ẍ+9 =. Hence cos3t)+c 2 sin3t) and so y = c 2 3 c )cos3t)+ 3 c 2 + c )sin3t). We may write the answer in the vector form as a linear combination of two independent solutions: cos3t) )+c cos3t)+ 3 sin3t) 2 ) sin3t) 3 cos3t)+ sin3t). Approach 2. Observe that 5 ẋ 2ẏ =. Rearranging terms, 5ẋ ẋy +ẏ)+2ẏy =, and integrating the result with respect to t, we obtain an implicit description of trajectories: 5 2 2y +2y 2 = C. This family of ellipses may be parameterized as above. Approach 3. Determine the eigendirections. For r = 3i, the eigenvector condition is 2 z z = 3i. 5 z 2 z 2 This gives a system of two dependent linear equations, { 3i)z 2z 2 = 5z +3i)z 2 =, satisfied by z 2 =. z 2 3i 2 Similarly, is an eigenvector corresponding to the eigenvalue r = 3i. We thus +3i have a pair of independent comple solutions e 3it, e 3i 3it. +3i Separating real and imaginary parts, we deduce the general solution: 2cos3t) 2sin3t) )+c cos3t) + 3 sin3t) 2. 3 cos3t) + sin3t) 4

5

Phase portraits in two dimensions

Phase portraits in two dimensions Phase portraits in two dimensions 8.3, Spring, 999 It [ is convenient to represent the solutions to an autonomous system x = f( x) (where x x = ) by means of a phase portrait. The x, y plane is called

More information

Even-Numbered Homework Solutions

Even-Numbered Homework Solutions -6 Even-Numbered Homework Solutions Suppose that the matric B has λ = + 5i as an eigenvalue with eigenvector Y 0 = solution to dy = BY Using Euler s formula, we can write the complex-valued solution Y

More information

Section 9.3 Phase Plane Portraits (for Planar Systems)

Section 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 information

Homogeneous Constant Matrix Systems, Part II

Homogeneous Constant Matrix Systems, Part II 4 Homogeneous Constant Matri Systems, Part II Let us now epand our discussions begun in the previous chapter, and consider homogeneous constant matri systems whose matrices either have comple eigenvalues

More information

Homogeneous Constant Matrix Systems, Part II

Homogeneous Constant Matrix Systems, Part II 4 Homogeneous Constant Matrix Systems, Part II Let us now expand our discussions begun in the previous chapter, and consider homogeneous constant matrix systems whose matrices either have complex eigenvalues

More information

21.55 Worksheet 7 - preparation problems - question 1:

21.55 Worksheet 7 - preparation problems - question 1: Dynamics 76. Worksheet 7 - preparation problems - question : A coupled oscillator with two masses m and positions x (t) and x (t) is described by the following equations of motion: ẍ x + 8x ẍ x +x A. Write

More information

Linearization of Differential Equation Models

Linearization of Differential Equation Models Linearization of Differential Equation Models 1 Motivation We cannot solve most nonlinear models, so we often instead try to get an overall feel for the way the model behaves: we sometimes talk about looking

More information

Autonomous system = system without inputs

Autonomous system = system without inputs Autonomous system = system without inputs State space representation B(A,C) = {y there is x, such that σx = Ax, y = Cx } x is the state, n := dim(x) is the state dimension, y is the output Polynomial representation

More information

Understand the existence and uniqueness theorems and what they tell you about solutions to initial value problems.

Understand 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 information

MATH 24 EXAM 3 SOLUTIONS

MATH 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 information

Math 1302, Week 8: Oscillations

Math 1302, Week 8: Oscillations Math 302, Week 8: Oscillations T y eq Y y = y eq + Y mg Figure : Simple harmonic motion. At equilibrium the string is of total length y eq. During the motion we let Y be the extension beyond equilibrium,

More information

Lecture 7. Please note. Additional tutorial. Please note that there is no lecture on Tuesday, 15 November 2011.

Lecture 7. Please note. Additional tutorial. Please note that there is no lecture on Tuesday, 15 November 2011. Lecture 7 3 Ordinary differential equations (ODEs) (continued) 6 Linear equations of second order 7 Systems of differential equations Please note Please note that there is no lecture on Tuesday, 15 November

More information

ELEC 3035, Lecture 3: Autonomous systems Ivan Markovsky

ELEC 3035, Lecture 3: Autonomous systems Ivan Markovsky ELEC 3035, Lecture 3: Autonomous systems Ivan Markovsky Equilibrium points and linearization Eigenvalue decomposition and modal form State transition matrix and matrix exponential Stability ELEC 3035 (Part

More information

Math 312 Lecture Notes Linear Two-dimensional Systems of Differential Equations

Math 312 Lecture Notes Linear Two-dimensional Systems of Differential Equations Math 2 Lecture Notes Linear Two-dimensional Systems of Differential Equations Warren Weckesser Department of Mathematics Colgate University February 2005 In these notes, we consider the linear system of

More information

STABILITY. Phase portraits and local stability

STABILITY. Phase portraits and local stability MAS271 Methods for differential equations Dr. R. Jain STABILITY Phase portraits and local stability We are interested in system of ordinary differential equations of the form ẋ = f(x, y), ẏ = g(x, y),

More information

Physics GRE Practice

Physics GRE Practice Physics GRE Practice Chapter 3: Harmonic Motion in 1-D Harmonic Motion occurs when the acceleration of a system is proportional to the negative of its displacement. or a x (1) ẍ x (2) Examples of Harmonic

More information

Chapter 14 (Oscillations) Key concept: Downloaded from

Chapter 14 (Oscillations) Key concept: Downloaded from Chapter 14 (Oscillations) Multiple Choice Questions Single Correct Answer Type Q1. The displacement of a particle is represented by the equation. The motion of the particle is (a) simple harmonic with

More information

Linear Differential Equations. Problems

Linear 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 information

154 Chapter 9 Hints, Answers, and Solutions The particular trajectories are highlighted in the phase portraits below.

154 Chapter 9 Hints, Answers, and Solutions The particular trajectories are highlighted in the phase portraits below. 54 Chapter 9 Hints, Answers, and Solutions 9. The Phase Plane 9.. 4. The particular trajectories are highlighted in the phase portraits below... 3. 4. 9..5. Shown below is one possibility with x(t) and

More information

Seminar 6: COUPLED HARMONIC OSCILLATORS

Seminar 6: COUPLED HARMONIC OSCILLATORS Seminar 6: COUPLED HARMONIC OSCILLATORS 1. Lagrangian Equations of Motion Let consider a system consisting of two harmonic oscillators that are coupled together. As a model, we will use two particles attached

More information

93 Analytical solution of differential equations

93 Analytical solution of differential equations 1 93 Analytical solution of differential equations 1. Nonlinear differential equation The only kind of nonlinear differential equations that we solve analytically is the so-called separable differential

More information

10. Operators and the Exponential Response Formula

10. Operators and the Exponential Response Formula 52 10. Operators and the Exponential Response Formula 10.1. Operators. Operators are to functions as functions are to numbers. An operator takes a function, does something to it, and returns this modified

More information

Math 216 Final Exam 24 April, 2017

Math 216 Final Exam 24 April, 2017 Math 216 Final Exam 24 April, 2017 This sample exam is provided to serve as one component of your studying for this exam in this course. Please note that it is not guaranteed to cover the material that

More information

Answers and Hints to Review Questions for Test (a) Find the general solution to the linear system of differential equations Y = 2 ± 3i.

Answers and Hints to Review Questions for Test (a) Find the general solution to the linear system of differential equations Y = 2 ± 3i. Answers and Hints to Review Questions for Test 3 (a) Find the general solution to the linear system of differential equations [ dy 3 Y 3 [ (b) Find the specific solution that satisfies Y (0) = (c) What

More information

Copyright (c) 2006 Warren Weckesser

Copyright (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 information

APPM 2360: Final Exam 10:30am 1:00pm, May 6, 2015.

APPM 2360: Final Exam 10:30am 1:00pm, May 6, 2015. APPM 23: Final Exam :3am :pm, May, 25. ON THE FRONT OF YOUR BLUEBOOK write: ) your name, 2) your student ID number, 3) lecture section, 4) your instructor s name, and 5) a grading table for eight questions.

More information

+ i. cos(t) + 2 sin(t) + c 2.

+ i. cos(t) + 2 sin(t) + c 2. MATH HOMEWORK #7 PART A SOLUTIONS Problem 7.6.. Consider the system x = 5 x. a Express the general solution of the given system of equations in terms of realvalued functions. b Draw a direction field,

More information

A plane autonomous system is a pair of simultaneous first-order differential equations,

A plane autonomous system is a pair of simultaneous first-order differential equations, Chapter 11 Phase-Plane Techniques 11.1 Plane Autonomous Systems A plane autonomous system is a pair of simultaneous first-order differential equations, ẋ = f(x, y), ẏ = g(x, y). This system has an equilibrium

More information

An Exactly Solvable 3 Body Problem

An Exactly Solvable 3 Body Problem An Exactly Solvable 3 Body Problem The most famous n-body problem is one where particles interact by an inverse square-law force. However, there is a class of exactly solvable n-body problems in which

More information

Def. (a, b) is a critical point of the autonomous system. 1 Proper node (stable or unstable) 2 Improper node (stable or unstable)

Def. (a, b) is a critical point of the autonomous system. 1 Proper node (stable or unstable) 2 Improper node (stable or unstable) Types of critical points Def. (a, b) is a critical point of the autonomous system Math 216 Differential Equations Kenneth Harris kaharri@umich.edu Department of Mathematics University of Michigan November

More information

20D - Homework Assignment 5

20D - 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 information

Physics 5153 Classical Mechanics. Canonical Transformations-1

Physics 5153 Classical Mechanics. Canonical Transformations-1 1 Introduction Physics 5153 Classical Mechanics Canonical Transformations The choice of generalized coordinates used to describe a physical system is completely arbitrary, but the Lagrangian is invariant

More information

DIAGONALIZABLE LINEAR SYSTEMS AND STABILITY. 1. Algebraic facts. We first recall two descriptions of matrix multiplication.

DIAGONALIZABLE LINEAR SYSTEMS AND STABILITY. 1. Algebraic facts. We first recall two descriptions of matrix multiplication. DIAGONALIZABLE LINEAR SYSTEMS AND STABILITY. 1. Algebraic facts. We first recall two descriptions of matrix multiplication. Let A be n n, P be n r, given by its columns: P = [v 1 v 2... v r ], where the

More information

Differential Equations 2280 Sample Midterm Exam 3 with Solutions Exam Date: 24 April 2015 at 12:50pm

Differential Equations 2280 Sample Midterm Exam 3 with Solutions Exam Date: 24 April 2015 at 12:50pm Differential Equations 228 Sample Midterm Exam 3 with Solutions Exam Date: 24 April 25 at 2:5pm Instructions: This in-class exam is 5 minutes. No calculators, notes, tables or books. No answer check is

More information

Linear Planar Systems Math 246, Spring 2009, Professor David Levermore We now consider linear systems of the form

Linear Planar Systems Math 246, Spring 2009, Professor David Levermore We now consider linear systems of the form Linear Planar Systems Math 246, Spring 2009, Professor David Levermore We now consider linear systems of the form d x x 1 = A, where A = dt y y a11 a 12 a 21 a 22 Here the entries of the coefficient matrix

More information

Assignment 6. Using the result for the Lagrangian for a double pendulum in Problem 1.22, we get

Assignment 6. Using the result for the Lagrangian for a double pendulum in Problem 1.22, we get Assignment 6 Goldstein 6.4 Obtain the normal modes of vibration for the double pendulum shown in Figure.4, assuming equal lengths, but not equal masses. Show that when the lower mass is small compared

More information

MathQuest: Differential Equations

MathQuest: Differential Equations MathQuest: Differential Equations Solutions to Linear Systems. Consider the linear system given by dy dt = 4 True or False: Y e t t = is a solution. c False, but I am not very confident Y.. Consider the

More information

Nonlinear Autonomous Systems of Differential

Nonlinear Autonomous Systems of Differential Chapter 4 Nonlinear Autonomous Systems of Differential Equations 4.0 The Phase Plane: Linear Systems 4.0.1 Introduction Consider a system of the form x = A(x), (4.0.1) where A is independent of t. Such

More information

4. Complex Oscillations

4. Complex Oscillations 4. Complex Oscillations The most common use of complex numbers in physics is for analyzing oscillations and waves. We will illustrate this with a simple but crucially important model, the damped harmonic

More information

Solutions to Math 53 Math 53 Practice Final

Solutions to Math 53 Math 53 Practice Final Solutions to Math 5 Math 5 Practice Final 20 points Consider the initial value problem y t 4yt = te t with y 0 = and y0 = 0 a 8 points Find the Laplace transform of the solution of this IVP b 8 points

More information

Section 8.2 : Homogeneous Linear Systems

Section 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 information

Hello everyone, Best, Josh

Hello everyone, Best, Josh Hello everyone, As promised, the chart mentioned in class about what kind of critical points you get with different types of eigenvalues are included on the following pages (The pages are an ecerpt from

More information

Normal Modes, Wave Motion and the Wave Equation Hilary Term 2011 Lecturer: F Hautmann

Normal Modes, Wave Motion and the Wave Equation Hilary Term 2011 Lecturer: F Hautmann Normal Modes, Wave Motion and the Wave Equation Hilary Term 2011 Lecturer: F Hautmann Part A: Normal modes ( 4 lectures) Part B: Waves ( 8 lectures) printed lecture notes slides will be posted on lecture

More information

APPM 2360 Section Exam 3 Wednesday November 19, 7:00pm 8:30pm, 2014

APPM 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 information

Solution Set Five. 2 Problem #2: The Pendulum of Doom Equation of Motion Simple Pendulum Limit Visualization...

Solution Set Five. 2 Problem #2: The Pendulum of Doom Equation of Motion Simple Pendulum Limit Visualization... : Solution Set Five Northwestern University, Classical Mechanics Classical Mechanics, Third Ed.- Goldstein November 4, 2015 Contents 1 Problem #1: Coupled Mass Oscillator System. 2 1.1 Normal Modes.......................................

More information

Calculus and Differential Equations II

Calculus and Differential Equations II MATH 250 B Second order autonomous linear systems We are mostly interested with 2 2 first order autonomous systems of the form { x = a x + b y y = c x + d y where x and y are functions of t and a, b, c,

More information

Limit Cycles II. Prof. Ned Wingreen MOL 410/510. How to prove a closed orbit exists?

Limit Cycles II. Prof. Ned Wingreen MOL 410/510. How to prove a closed orbit exists? Limit Cycles II Prof. Ned Wingreen MOL 410/510 How to prove a closed orbit eists? numerically Poincaré-Bendison Theorem: If 1. R is a closed, bounded subset of the plane. = f( ) is a continuously differentiable

More information

MathQuest: Differential Equations

MathQuest: Differential Equations MathQuest: Differential Equations Geometry of Systems 1. The differential equation d Y dt = A Y has two straight line solutions corresponding to [ ] [ ] 1 1 eigenvectors v 1 = and v 2 2 = that are shown

More information

Solutions of Spring 2008 Final Exam

Solutions of Spring 2008 Final Exam Solutions of Spring 008 Final Exam 1. (a) The isocline for slope 0 is the pair of straight lines y = ±x. The direction field along these lines is flat. The isocline for slope is the hyperbola on the left

More information

Solutions a) The characteristic equation is r = 0 with roots ±2i, so the complementary solution is. y c = c 1 cos(2t)+c 2 sin(2t).

Solutions a) The characteristic equation is r = 0 with roots ±2i, so the complementary solution is. y c = c 1 cos(2t)+c 2 sin(2t). Solutions 3.9. a) The characteristic equation is r + 4 = 0 with roots ±i, so the complementar solution is c = c cos(t)+c sin(t). b) We look for a particular solution in the form and obtain the equations

More information

Phys 7221 Homework # 8

Phys 7221 Homework # 8 Phys 71 Homework # 8 Gabriela González November 15, 6 Derivation 5-6: Torque free symmetric top In a torque free, symmetric top, with I x = I y = I, the angular velocity vector ω in body coordinates with

More information

Chapter 2: Complex numbers

Chapter 2: Complex numbers Chapter 2: Complex numbers Complex numbers are commonplace in physics and engineering. In particular, complex numbers enable us to simplify equations and/or more easily find solutions to equations. We

More information

3 + 4i 2 + 3i. 3 4i Fig 1b

3 + 4i 2 + 3i. 3 4i Fig 1b The introduction of complex numbers in the 16th century was a natural step in a sequence of extensions of the positive integers, starting with the introduction of negative numbers (to solve equations of

More information

Math 142-2, Homework 1

Math 142-2, Homework 1 Math 142-2, Homework 1 Your name here Problem 5.8 (a) Show that x = c 1 cosωt+c 2 sinωt is the general solution of mx = kx. What is the value of ω? (b) Show that an equivalent expression for the general

More information

ENGI 9420 Lecture Notes 4 - Stability Analysis Page Stability Analysis for Non-linear Ordinary Differential Equations

ENGI 9420 Lecture Notes 4 - Stability Analysis Page Stability Analysis for Non-linear Ordinary Differential Equations ENGI 940 Lecture Notes 4 - Stability Analysis Page 4.01 4. Stability Analysis for Non-linear Ordinary Differential Equations A pair of simultaneous first order homogeneous linear ordinary differential

More information

ODE Homework 1. Due Wed. 19 August 2009; At the beginning of the class

ODE Homework 1. Due Wed. 19 August 2009; At the beginning of the class ODE Homework Due Wed. 9 August 2009; At the beginning of the class. (a) Solve Lẏ + Ry = E sin(ωt) with y(0) = k () L, R, E, ω are positive constants. (b) What is the limit of the solution as ω 0? (c) Is

More information

Complex Dynamic Systems: Qualitative vs Quantitative analysis

Complex Dynamic Systems: Qualitative vs Quantitative analysis Complex Dynamic Systems: Qualitative vs Quantitative analysis Complex Dynamic Systems Chiara Mocenni Department of Information Engineering and Mathematics University of Siena (mocenni@diism.unisi.it) Dynamic

More information

Symmetries 2 - Rotations in Space

Symmetries 2 - Rotations in Space Symmetries 2 - Rotations in Space This symmetry is about the isotropy of space, i.e. space is the same in all orientations. Thus, if we continuously rotated an entire system in space, we expect the system

More information

Classical Mechanics Comprehensive Exam Solution

Classical Mechanics Comprehensive Exam Solution Classical Mechanics Comprehensive Exam Solution January 31, 011, 1:00 pm 5:pm Solve the following six problems. In the following problems, e x, e y, and e z are unit vectors in the x, y, and z directions,

More information

Appendix: A Computer-Generated Portrait Gallery

Appendix: A Computer-Generated Portrait Gallery Appendi: A Computer-Generated Portrait Galler There are a number of public-domain computer programs which produce phase portraits for 2 2 autonomous sstems. One has the option of displaing the trajectories

More information

Background ODEs (2A) Young Won Lim 3/7/15

Background ODEs (2A) Young Won Lim 3/7/15 Background ODEs (2A) Copyright (c) 2014-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any

More information

Math 304 Answers to Selected Problems

Math 304 Answers to Selected Problems Math Answers to Selected Problems Section 6.. Find the general solution to each of the following systems. a y y + y y y + y e y y y y y + y f y y + y y y + 6y y y + y Answer: a This is a system of the

More information

Math 216 Second Midterm 20 March, 2017

Math 216 Second Midterm 20 March, 2017 Math 216 Second Midterm 20 March, 2017 This sample exam is provided to serve as one component of your studying for this exam in this course. Please note that it is not guaranteed to cover the material

More information

21 Linear State-Space Representations

21 Linear State-Space Representations ME 132, Spring 25, UC Berkeley, A Packard 187 21 Linear State-Space Representations First, let s describe the most general type of dynamic system that we will consider/encounter in this class Systems may

More information

Systems of Second Order Differential Equations Cayley-Hamilton-Ziebur

Systems of Second Order Differential Equations Cayley-Hamilton-Ziebur Systems of Second Order Differential Equations Cayley-Hamilton-Ziebur Characteristic Equation Cayley-Hamilton Cayley-Hamilton Theorem An Example Euler s Substitution for u = A u The Cayley-Hamilton-Ziebur

More information

Math 216 Final Exam 24 April, 2017

Math 216 Final Exam 24 April, 2017 Math 216 Final Exam 24 April, 2017 This sample exam is provided to serve as one component of your studying for this exam in this course. Please note that it is not guaranteed to cover the material that

More information

Notes on the Periodically Forced Harmonic Oscillator

Notes on the Periodically Forced Harmonic Oscillator Notes on the Periodically orced Harmonic Oscillator Warren Weckesser Math 38 - Differential Equations 1 The Periodically orced Harmonic Oscillator. By periodically forced harmonic oscillator, we mean the

More information

Consider a particle in 1D at position x(t), subject to a force F (x), so that mẍ = F (x). Define the kinetic energy to be.

Consider a particle in 1D at position x(t), subject to a force F (x), so that mẍ = F (x). Define the kinetic energy to be. Chapter 4 Energy and Stability 4.1 Energy in 1D Consider a particle in 1D at position x(t), subject to a force F (x), so that mẍ = F (x). Define the kinetic energy to be T = 1 2 mẋ2 and the potential energy

More information

Dynamical Systems Solutions to Exercises

Dynamical Systems Solutions to Exercises Dynamical Systems Part 5-6 Dr G Bowtell Dynamical Systems Solutions to Exercises. Figure : Phase diagrams for i, ii and iii respectively. Only fixed point is at the origin since the equations are linear

More information

Math K (24564) - Lectures 02

Math K (24564) - Lectures 02 Math 39100 K (24564) - Lectures 02 Ethan Akin Office: NAC 6/287 Phone: 650-5136 Email: ethanakin@earthlink.net Spring, 2018 Contents Second Order Linear Equations, B & D Chapter 4 Second Order Linear Homogeneous

More information

Chapter 6 Nonlinear Systems and Phenomena. Friday, November 2, 12

Chapter 6 Nonlinear Systems and Phenomena. Friday, November 2, 12 Chapter 6 Nonlinear Systems and Phenomena 6.1 Stability and the Phase Plane We now move to nonlinear systems Begin with the first-order system for x(t) d dt x = f(x,t), x(0) = x 0 In particular, consider

More information

Introduction to Dynamical Systems. Tuesday, September 9, 14

Introduction to Dynamical Systems. Tuesday, September 9, 14 Introduction to Dynamical Systems 1 Dynamical Systems A dynamical system is a set of related phenomena that change over time in a deterministic way. The future states of the system can be predicted from

More information

PHYSICS 3204 PUBLIC EXAM QUESTIONS (Magnetism &Electromagnetism)

PHYSICS 3204 PUBLIC EXAM QUESTIONS (Magnetism &Electromagnetism) PHYSICS 3204 PUBLIC EXAM QUESTIONS (Magnetism &Electromagnetism) NAME: August 2009---------------------------------------------------------------------------------------------------------------------------------

More information

Waves & Normal Modes. Matt Jarvis

Waves & Normal Modes. Matt Jarvis Waves & Normal Modes Matt Jarvis February 27, 2016 Contents 1 Oscillations 2 1.0.1 Simple Harmonic Motion - revision................... 2 2 Normal Modes 5 2.1 The coupled pendulum..............................

More information

Econ 204 Differential Equations. 1 Existence and Uniqueness of Solutions

Econ 204 Differential Equations. 1 Existence and Uniqueness of Solutions Econ 4 Differential Equations In this supplement, we use the methods we have developed so far to study differential equations. 1 Existence and Uniqueness of Solutions Definition 1 A differential equation

More information

16.30/31, Fall 2010 Recitation # 13

16.30/31, Fall 2010 Recitation # 13 16.30/31, Fall 2010 Recitation # 13 Brandon Luders December 6, 2010 In this recitation, we tie the ideas of Lyapunov stability analysis (LSA) back to previous ways we have demonstrated stability - but

More information

Systems of Algebraic Equations and Systems of Differential Equations

Systems of Algebraic Equations and Systems of Differential Equations Systems of Algebraic Equations and Systems of Differential Equations Topics: 2 by 2 systems of linear equations Matrix expression; Ax = b Solving 2 by 2 homogeneous systems Functions defined on matrices

More information

L = 1 2 a(q) q2 V (q).

L = 1 2 a(q) q2 V (q). Physics 3550, Fall 2011 Motion near equilibrium - Small Oscillations Relevant Sections in Text: 5.1 5.6 Motion near equilibrium 1 degree of freedom One of the most important situations in physics is motion

More information

Final Exam December 20, 2011

Final Exam December 20, 2011 Final Exam December 20, 2011 Math 420 - Ordinary Differential Equations No credit will be given for answers without mathematical or logical justification. Simplify answers as much as possible. Leave solutions

More information

June 2011 PURDUE UNIVERSITY Study Guide for the Credit Exam in (MA 262) Linear Algebra and Differential Equations

June 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 information

Solutions to Dynamical Systems 2010 exam. Each question is worth 25 marks.

Solutions to Dynamical Systems 2010 exam. Each question is worth 25 marks. Solutions to Dynamical Systems exam Each question is worth marks [Unseen] Consider the following st order differential equation: dy dt Xy yy 4 a Find and classify all the fixed points of Hence draw the

More information

MTH 464: Computational Linear Algebra

MTH 464: Computational Linear Algebra MTH 464: Computational Linear Algebra Lecture Outlines Exam 4 Material Prof. M. Beauregard Department of Mathematics & Statistics Stephen F. Austin State University April 15, 2018 Linear Algebra (MTH 464)

More information

3 Stability and Lyapunov Functions

3 Stability and Lyapunov Functions CDS140a Nonlinear Systems: Local Theory 02/01/2011 3 Stability and Lyapunov Functions 3.1 Lyapunov Stability Denition: An equilibrium point x 0 of (1) is stable if for all ɛ > 0, there exists a δ > 0 such

More information

Applications of Diagonalization

Applications of Diagonalization Applications of Diagonalization Hsiu-Hau Lin hsiuhau@phys.nthu.edu.tw Apr 2, 200 The notes cover applications of matrix diagonalization Boas 3.2. Quadratic curves Consider the quadratic curve, 5x 2 4xy

More information

1 Simple Harmonic Oscillator

1 Simple Harmonic Oscillator Massachusetts Institute of Technology MITES 2017 Physics III Lectures 02 and 03: Simple Harmonic Oscillator, Classical Pendulum, and General Oscillations In these notes, we introduce simple harmonic oscillator

More information

Modal Decomposition and the Time-Domain Response of Linear Systems 1

Modal 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 information

Fundamentals of Dynamical Systems / Discrete-Time Models. Dr. Dylan McNamara people.uncw.edu/ mcnamarad

Fundamentals of Dynamical Systems / Discrete-Time Models. Dr. Dylan McNamara people.uncw.edu/ mcnamarad Fundamentals of Dynamical Systems / Discrete-Time Models Dr. Dylan McNamara people.uncw.edu/ mcnamarad Dynamical systems theory Considers how systems autonomously change along time Ranges from Newtonian

More information

M2A2 Problem Sheet 3 - Hamiltonian Mechanics

M2A2 Problem Sheet 3 - Hamiltonian Mechanics MA Problem Sheet 3 - Hamiltonian Mechanics. The particle in a cone. A particle slides under gravity, inside a smooth circular cone with a vertical axis, z = k x + y. Write down its Lagrangian in a) Cartesian,

More information

Week 9 solutions. k = mg/l = /5 = 3920 g/s 2. 20u + 400u u = 0,

Week 9 solutions. k = mg/l = /5 = 3920 g/s 2. 20u + 400u u = 0, Week 9 solutions ASSIGNMENT 20. (Assignment 19 had no hand-graded component.) 3.7.9. A mass of 20 g stretches a spring 5 cm. Suppose that the mass is also attached to a viscous damper with a damping constant

More information

Nonlinear dynamics & chaos BECS

Nonlinear dynamics & chaos BECS Nonlinear dynamics & chaos BECS-114.7151 Phase portraits Focus: nonlinear systems in two dimensions General form of a vector field on the phase plane: Vector notation: Phase portraits Solution x(t) describes

More information

u n 2 4 u n 36 u n 1, n 1.

u n 2 4 u n 36 u n 1, n 1. Exercise 1 Let (u n ) be the sequence defined by Set v n = u n 1 x+ u n and f (x) = 4 x. 1. Solve the equations f (x) = 1 and f (x) =. u 0 = 0, n Z +, u n+1 = u n + 4 u n.. Prove that if u n < 1, then

More information

ANSWERS Final Exam Math 250b, Section 2 (Professor J. M. Cushing), 15 May 2008 PART 1

ANSWERS 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 information

Introductory Physics. Week 2015/05/29

Introductory Physics. Week 2015/05/29 2015/05/29 Part I Summary of week 6 Summary of week 6 We studied the motion of a projectile under uniform gravity, and constrained rectilinear motion, introducing the concept of constraint force. Then

More information

ENGI Linear Approximation (2) Page Linear Approximation to a System of Non-Linear ODEs (2)

ENGI Linear Approximation (2) Page Linear Approximation to a System of Non-Linear ODEs (2) ENGI 940 4.06 - Linear Approximation () Page 4. 4.06 Linear Approximation to a System of Non-Linear ODEs () From sections 4.0 and 4.0, the non-linear system dx dy = x = P( x, y), = y = Q( x, y) () with

More information

G : Statistical Mechanics

G : Statistical Mechanics G5.651: Statistical Mechanics Notes for Lecture 1 I. DERIVATION OF THE DISCRETIZED PATH INTEGRAL We begin our discussion of the Feynman path integral with the canonical ensemble. The epressions for the

More information

Third In-Class Exam Solutions Math 246, Professor David Levermore Thursday, 3 December 2009 (1) [6] Given that 2 is an eigenvalue of the matrix

Third In-Class Exam Solutions Math 246, Professor David Levermore Thursday, 3 December 2009 (1) [6] Given that 2 is an eigenvalue of the matrix Third In-Class Exam Solutions Math 26, Professor David Levermore Thursday, December 2009 ) [6] Given that 2 is an eigenvalue of the matrix A 2, 0 find all the eigenvectors of A associated with 2. Solution.

More information

5 Nonlinear equations

5 Nonlinear equations Approach 5 Nonlinear equations 5. Approach It is often difficult or impossible to obtain explicit solutions to higher order or coupled systems of nonlinear ordinary differential equations. The techniques

More information

FINAL EXAM MAY 20, 2004

FINAL EXAM MAY 20, 2004 18.034 FINAL EXAM MAY 20, 2004 Name: Problem 1: /10 Problem 2: /20 Problem 3: /25 Problem 4: /15 Problem 5: /20 Problem 6: /25 Problem 7: /10 Problem 8: /35 Problem 9: /40 Problem 10: /10 Extra credit

More information

Partial Differential Equations for Engineering Math 312, Fall 2012

Partial Differential Equations for Engineering Math 312, Fall 2012 Partial Differential Equations for Engineering Math 312, Fall 2012 Jens Lorenz July 17, 2012 Contents Department of Mathematics and Statistics, UNM, Albuquerque, NM 87131 1 Second Order ODEs with Constant

More information

AA 242B / ME 242B: Mechanical Vibrations (Spring 2016)

AA 242B / ME 242B: Mechanical Vibrations (Spring 2016) AA 242B / ME 242B: Mechanical Vibrations (Spring 206) Solution of Homework #3 Control Tab Figure : Schematic for the control tab. Inadequacy of a static-test A static-test for measuring θ would ideally

More information