1 The relation between a second order linear ode and a system of two rst order linear odes

Size: px
Start display at page:

Download "1 The relation between a second order linear ode and a system of two rst order linear odes"

Transcription

1 Math 1280 Spring, The relation between a second order linear ode and a system of two rst order linear odes In Chapter 3 of the text you learn to solve some second order linear ode's, such as x 00 + x = 0 (1) Then, in chapter 7, you studied systems of two rst order linear ode's, such as y 0 = x (2) In fact, you saw that (1) and (2) are completely equivalent to each other. Suppose, for example, that (x (t) ; y (t)) solves (2). Then by dierentiating the rst equation of (2), we get x 00 = y 0 Then use the second equation of (2) to get x 00 = x; or x 00 + x = 0 Similarly, if x (t) solves (1) ; and we set y (t) = x 0 ; then the pair of functions (x (t) ; y (t)) solves the system (2) This can be used to help draw phase planes. Consider the system y 0 = x 2y (3) Dierentiating the rst equation and using the second, we get or You learned to solve this previously. x 00 = y 0 = x 2y = x 2x 0 ; x x + x = 0 Substituting x = e rt ; we get r 2 + 2r + 1 = 0 (r + 1) 2 = 0; so r 1 = r 2 = 1 Therefore one solution is x 1 (t) = e t. You learned in chapter 3 that a second solution is given by x 2 (t) = te t 1

2 Converting to a system, we have x 0 1 (t) = e t = y 1 (t) Hence one solution to (3) is (x 1 (t) ; y 1 (t)) = e t ; e t For a second solution, we have x 2 (t) = te t ; x 0 2 (t) = e t te t = y 2 (t). Thus, a second solution of (3) is (x 2 (t) ; y 2 (t)) = te t ; e t te t We now wish to graph each of these in the phase plane. The rst is easy, because we easily see that y = x The second is not so easy to see, but the computer can do it easily. The following graph contains both trajectories. The following graph lls in some more of the trajectories. These will be further explained in class. See also case (3b) in section 9.1 of the text. Note that the matrix here is It turns out that the eigenvalues are r 1 = r 2 = 1; and there is only one linearly 1 independent eigenvector, say Observe that the two trajectories which are 1 straight lines above are on the line pointing in the direction of this vector. (One in x < 0; the other in x > 0) 2

3 2 Nonlinear phase planes Usually, a system x 0 = f (x; y) y 0 = g (x; y) is nonlinear, and can't be solved. We will learn various techniques for guring out the phase plane. The rst step is almost always to look for the equilibrium points. (The text calls these \critical points".) This is a matter of solving some equations, and can be tricky. Go over example 1 on page 502. Here is another example y 0 = y + x x 3 From the rst equation we see that y = 0 at any equilibrium point, and from the second, we see that there are three equilibrium points (0; 0), (1; 0) ; and ( 1; 0). Now we use PPlane to try to understand the phase plane. In doing so, we must be sure that the \window" includes the equilibrium points. We can use tools from the menu at the top of the phase plane window to show the equilibrium points clearly, and to choose a useful window. You may nd other useful tools by browsing along the top set of buttons. We will discuss the results in class. In the next section we discuss a very important analytical method for analyzing the phase plane for nonlinear systems. 3 The linearization of a nonlinear system around an equilibrium point In this section I will use the alternative notation for partial derivatives f x (x; y) (x; Consider a general system of two autonomous equations in two unknowns x 0 = f (x; y) y 0 = g (x; y) (4) 3

4 Suppose that (x 0 ; y 0 ) is an equilibrium point for (4). Then the \linearization of (4) around (x 0 ; y 0 ) is the linear system where u 0 = Au fx (x 0 ; y 0 ) f y (x 0 ; y 0 ) g x (x 0 ; y 0 ) g y (x 0 ; y 0 ) Notice that I use a dierent letter for the unknown function for the linearized system. The text does this also { see Example 3 in section 9.3. Here is a dierent example Clearly, in this example, y 0 = x + x 3 (5) f (x; y) = y g (x; y) = x + x 3 First we look for equilibrium points. It is easy to see that the only equilibrium point is (0; 0) We then nd the needed partial derivatives f x (x; y) = 0; f y (x; y) = 1 g x (x; y) = 1 + 3x 3 ; g y (x; y) = 0 The matrix A is therefore and the linearized system is ; u 0 = v v 0 = u Note that the solutions of the linear system do not involve sine and cosine. Instead, the general solution is e t e t u (t) = c 1 e t + c 2 e t (6) From this we can draw the phase plane. 4

5 We now wish to see what the relation between this picture is and the phase plane for the original system, (5). The picture below can easily be seen using PPlane. Can we see any similarity to the phase plane above? Notice that the \window" here is 10 x 10; 10 y 10 Let's zoom in. Now we can see a greater similarity. The window is 1 x 1; 1 y 1 We say that the linearized system gives \local" information about the phase plane, in some small region around the equilibrium point. The concept of linearization is one of the most important in applied mathematics. We will look at quite a few examples. Here is another one. y 0 = x y 3 (7) The only equilibrium point is (x 0 ; y 0 ) = (0; 0). We linearize around this point f x (x y) = 0; f y (x; y) = 1 g x (x; y) = 1; g y (x; y) = 3y 2 5

6 The linearized system is familiar We did the phase plane in class u 0 = v v 0 = u This is a \center". Now let's do the original system. (Will be passed out in class.) We nd spirals, not circles. In this case, it turns out, the linearized system is not a good predictor of the actual behavior of the system. We have to decide when it is useful, and when it is not. There is a very useful way of showing that the system (7) is a spiral and not a center. Recall that for the linear system, we know that u 2 + v 2 = c We can try to determine if, for (7), x 2 + y 2 is also constant. We do this by dierentiating and using the dierential equations. We get d dt x (t) 2 + y (t) 2 = 2xx 0 + 2yy 0 = 2xy + 2y x y 3 = 2y 4 This is negative, except when y = 0 So the quantity x (t) 2 + y (t) 2 is decreasing. Since x 2 +y 2 is the square of the distance from (x; y) to (0; 0), this shows that we get a stable spiral, and not a center for this system. (Well, some more has to be said, but I'll say that in class.) 6

7 4 Homework Due at the beginning of class on Wednesday, January 20. section 9.1, 9 section 9.2, 5, 7 section 9.3, 5, 10 7

Math 1270 Honors ODE I Fall, 2008 Class notes # 14. x 0 = F (x; y) y 0 = G (x; y) u 0 = au + bv = cu + dv

Math 1270 Honors ODE I Fall, 2008 Class notes # 14. x 0 = F (x; y) y 0 = G (x; y) u 0 = au + bv = cu + dv Math 1270 Honors ODE I Fall, 2008 Class notes # 1 We have learned how to study nonlinear systems x 0 = F (x; y) y 0 = G (x; y) (1) by linearizing around equilibrium points. If (x 0 ; y 0 ) is an equilibrium

More information

1 The pendulum equation

1 The pendulum equation Math 270 Honors ODE I Fall, 2008 Class notes # 5 A longer than usual homework assignment is at the end. The pendulum equation We now come to a particularly important example, the equation for an oscillating

More information

Math 1280 Notes 4 Last section revised, 1/31, 9:30 pm.

Math 1280 Notes 4 Last section revised, 1/31, 9:30 pm. 1 competing species Math 1280 Notes 4 Last section revised, 1/31, 9:30 pm. This section and the next deal with the subject of population biology. You will already have seen examples of this. Most calculus

More information

Math 312 Lecture Notes Linearization

Math 312 Lecture Notes Linearization Math 3 Lecture Notes Linearization Warren Weckesser Department of Mathematics Colgate University 3 March 005 These notes discuss linearization, in which a linear system is used to approximate the behavior

More information

Section 5.4 (Systems of Linear Differential Equation); 9.5 Eigenvalues and Eigenvectors, cont d

Section 5.4 (Systems of Linear Differential Equation); 9.5 Eigenvalues and Eigenvectors, cont d Section 5.4 (Systems of Linear Differential Equation); 9.5 Eigenvalues and Eigenvectors, cont d July 6, 2009 Today s Session Today s Session A Summary of This Session: Today s Session A Summary of This

More information

Lecture 10. (2) Functions of two variables. Partial derivatives. Dan Nichols February 27, 2018

Lecture 10. (2) Functions of two variables. Partial derivatives. Dan Nichols February 27, 2018 Lecture 10 Partial derivatives Dan Nichols nichols@math.umass.edu MATH 233, Spring 2018 University of Massachusetts February 27, 2018 Last time: functions of two variables f(x, y) x and y are the independent

More information

Second Order Linear Equations

Second 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 information

Exam 2 Study Guide: MATH 2080: Summer I 2016

Exam 2 Study Guide: MATH 2080: Summer I 2016 Exam Study Guide: MATH 080: Summer I 016 Dr. Peterson June 7 016 First Order Problems Solve the following IVP s by inspection (i.e. guessing). Sketch a careful graph of each solution. (a) u u; u(0) 0.

More information

Using web-based Java pplane applet to graph solutions of systems of differential equations

Using web-based Java pplane applet to graph solutions of systems of differential equations Using web-based Java pplane applet to graph solutions of systems of differential equations Our class project for MA 341 involves using computer tools to analyse solutions of differential equations. This

More information

Chapter 1, Section 1.2, Example 9 (page 13) and Exercise 29 (page 15). Use the Uniqueness Tool. Select the option ẋ = x

Chapter 1, Section 1.2, Example 9 (page 13) and Exercise 29 (page 15). Use the Uniqueness Tool. Select the option ẋ = x Use of Tools from Interactive Differential Equations with the texts Fundamentals of Differential Equations, 5th edition and Fundamentals of Differential Equations and Boundary Value Problems, 3rd edition

More information

Math 310 Introduction to Ordinary Differential Equations Final Examination August 9, Instructor: John Stockie

Math 310 Introduction to Ordinary Differential Equations Final Examination August 9, Instructor: John Stockie Make sure this exam has 15 pages. Math 310 Introduction to Ordinary Differential Equations inal Examination August 9, 2006 Instructor: John Stockie Name: (Please Print) Student Number: Special Instructions

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

MAT 22B - Lecture Notes

MAT 22B - Lecture Notes MAT 22B - Lecture Notes 4 September 205 Solving Systems of ODE Last time we talked a bit about how systems of ODE arise and why they are nice for visualization. Now we'll talk about the basics of how to

More information

Problem set 7 Math 207A, Fall 2011 Solutions

Problem set 7 Math 207A, Fall 2011 Solutions Problem set 7 Math 207A, Fall 2011 s 1. Classify the equilibrium (x, y) = (0, 0) of the system x t = x, y t = y + x 2. Is the equilibrium hyperbolic? Find an equation for the trajectories in (x, y)- phase

More information

3.3. SYSTEMS OF ODES 1. y 0 " 2y" y 0 + 2y = x1. x2 x3. x = y(t) = c 1 e t + c 2 e t + c 3 e 2t. _x = A x + f; x(0) = x 0.

3.3. SYSTEMS OF ODES 1. y 0  2y y 0 + 2y = x1. x2 x3. x = y(t) = c 1 e t + c 2 e t + c 3 e 2t. _x = A x + f; x(0) = x 0. .. SYSTEMS OF ODES. Systems of ODEs MATH 94 FALL 98 PRELIM # 94FA8PQ.tex.. a) Convert the third order dierential equation into a rst oder system _x = A x, with y " y" y + y = x = @ x x x b) The equation

More information

Linear Systems of ODE: Nullclines, Eigenvector lines and trajectories

Linear Systems of ODE: Nullclines, Eigenvector lines and trajectories Linear Systems of ODE: Nullclines, Eigenvector lines and trajectories James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 6, 203 Outline

More information

Lecture 9. Systems of Two First Order Linear ODEs

Lecture 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 information

1 A complete Fourier series solution

1 A complete Fourier series solution Math 128 Notes 13 In this last set of notes I will try to tie up some loose ends. 1 A complete Fourier series solution First here is an example of the full solution of a pde by Fourier series. Consider

More information

= 2e t e 2t + ( e 2t )e 3t = 2e t e t = e t. Math 20D Final Review

= 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 information

2.10 Saddles, Nodes, Foci and Centers

2.10 Saddles, Nodes, Foci and Centers 2.10 Saddles, Nodes, Foci and Centers In Section 1.5, a linear system (1 where x R 2 was said to have a saddle, node, focus or center at the origin if its phase portrait was linearly equivalent to one

More information

Linear Systems of ODE: Nullclines, Eigenvector lines and trajectories

Linear Systems of ODE: Nullclines, Eigenvector lines and trajectories Linear Systems of ODE: Nullclines, Eigenvector lines and trajectories James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 6, 2013 Outline

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

Math 1241, Spring 2014 Section 3.3. Rates of Change Average vs. Instantaneous Rates

Math 1241, Spring 2014 Section 3.3. Rates of Change Average vs. Instantaneous Rates Math 1241, Spring 2014 Section 3.3 Rates of Change Average vs. Instantaneous Rates Average Speed The concept of speed (distance traveled divided by time traveled) is a familiar instance of a rate of change.

More information

We will begin by first solving this equation on a rectangle in 2 dimensions with prescribed boundary data at each edge.

We will begin by first solving this equation on a rectangle in 2 dimensions with prescribed boundary data at each edge. Page 1 Sunday, May 31, 2015 9:24 PM From our study of the 2-d and 3-d heat equation in thermal equlibrium another PDE which we will learn to solve. Namely Laplace's Equation we arrive at In 3-d In 2-d

More information

Lab 5: Nonlinear Systems

Lab 5: Nonlinear Systems Lab 5: Nonlinear Systems Goals In this lab you will use the pplane6 program to study two nonlinear systems by direct numerical simulation. The first model, from population biology, displays interesting

More information

y x 3. Solve each of the given initial value problems. (a) y 0? xy = x, y(0) = We multiply the equation by e?x, and obtain Integrating both sides with

y x 3. Solve each of the given initial value problems. (a) y 0? xy = x, y(0) = We multiply the equation by e?x, and obtain Integrating both sides with Solutions to the Practice Problems Math 80 Febuary, 004. For each of the following dierential equations, decide whether the given function is a solution. (a) y 0 = (x + )(y? ), y =? +exp(x +x)?exp(x +x)

More 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

True or False. Circle T if the statement is always true; otherwise circle F. for all angles θ. T F. 1 sin θ

True or False. Circle T if the statement is always true; otherwise circle F. for all angles θ. T F. 1 sin θ Math 90 Practice Midterm III Solutions Ch. 8-0 (Ebersole), 3.3-3.8 (Stewart) DISCLAIMER. This collection of practice problems is not guaranteed to be identical, in length or content, to the actual exam.

More information

Vector Functions & Space Curves MATH 2110Q

Vector Functions & Space Curves MATH 2110Q Vector Functions & Space Curves Vector Functions & Space Curves Vector Functions Definition A vector function or vector-valued function is a function that takes real numbers as inputs and gives vectors

More information

Homework 6 Solutions. Solution. Note {e t, te t, t 2 e t, e 2t } is linearly independent. If β = {e t, te t, t 2 e t, e 2t }, then

Homework 6 Solutions. Solution. Note {e t, te t, t 2 e t, e 2t } is linearly independent. If β = {e t, te t, t 2 e t, e 2t }, then Homework 6 Solutions 1 Let V be the real vector space spanned by the functions e t, te t, t 2 e t, e 2t Find a Jordan canonical basis and a Jordan canonical form of T on V dened by T (f) = f Solution Note

More information

0 as an eigenvalue. degenerate

0 as an eigenvalue. degenerate Math 1 Topics since the third exam Chapter 9: Non-linear Sstems of equations x1: Tpical Phase Portraits The structure of the solutions to a linear, constant coefficient, sstem of differential equations

More information

Math 216 First Midterm 18 October, 2018

Math 216 First Midterm 18 October, 2018 Math 16 First Midterm 18 October, 018 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

1 Continuation of solutions

1 Continuation of solutions Math 175 Honors ODE I Spring 13 Notes 4 1 Continuation of solutions Theorem 1 in the previous notes states that a solution to y = f (t; y) (1) y () = () exists on a closed interval [ h; h] ; under certain

More information

Math 53 Spring 2018 Practice Midterm 2

Math 53 Spring 2018 Practice Midterm 2 Math 53 Spring 218 Practice Midterm 2 Nikhil Srivastava 8 minutes, closed book, closed notes 1. alculate 1 y 2 (x 2 + y 2 ) 218 dxdy Solution. Since the type 2 region D = { y 1, x 1 y 2 } is a quarter

More information

Additional Homework Problems

Additional Homework Problems Additional Homework Problems These problems supplement the ones assigned from the text. Use complete sentences whenever appropriate. Use mathematical terms appropriately. 1. What is the order of a differential

More information

Dynamical Systems. August 13, 2013

Dynamical Systems. August 13, 2013 Dynamical Systems Joshua Wilde, revised by Isabel Tecu, Takeshi Suzuki and María José Boccardi August 13, 2013 Dynamical Systems are systems, described by one or more equations, that evolve over time.

More information

Systems of Linear ODEs

Systems of Linear ODEs P a g e 1 Systems of Linear ODEs Systems of ordinary differential equations can be solved in much the same way as discrete dynamical systems if the differential equations are linear. We will focus here

More information

MAT 122 Homework 7 Solutions

MAT 122 Homework 7 Solutions MAT 1 Homework 7 Solutions Section 3.3, Problem 4 For the function w = (t + 1) 100, we take the inside function to be z = t + 1 and the outside function to be z 100. The derivative of the inside function

More information

Homework Solutions:

Homework Solutions: Homework Solutions: 1.1-1.3 Section 1.1: 1. Problems 1, 3, 5 In these problems, we want to compare and contrast the direction fields for the given (autonomous) differential equations of the form y = ay

More information

Wheels Radius / Distance Traveled

Wheels Radius / Distance Traveled Mechanics Teacher Note to the teacher On these pages, students will learn about the relationships between wheel radius, diameter, circumference, revolutions and distance. Students will use formulas relating

More information

McGill University April 16, Advanced Calculus for Engineers

McGill University April 16, Advanced Calculus for Engineers McGill University April 16, 2014 Faculty of cience Final examination Advanced Calculus for Engineers Math 264 April 16, 2014 Time: 6PM-9PM Examiner: Prof. R. Choksi Associate Examiner: Prof. A. Hundemer

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

Math 308 Final Exam Practice Problems

Math 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 information

Chapter 9 Global Nonlinear Techniques

Chapter 9 Global Nonlinear Techniques Chapter 9 Global Nonlinear Techniques Consider nonlinear dynamical system 0 Nullcline X 0 = F (X) = B @ f 1 (X) f 2 (X). f n (X) x j nullcline = fx : f j (X) = 0g equilibrium solutions = intersection of

More information

*****DO NOT WRITE ON THIS PAPER***** Answer these questions in your SPIRAL NOTEBOOK. # 9 BUILD AN ATOM: Post-lab questions

*****DO NOT WRITE ON THIS PAPER***** Answer these questions in your SPIRAL NOTEBOOK. # 9 BUILD AN ATOM: Post-lab questions *****DO NOT WRITE ON THIS PAPER***** Answer these questions in your SPIRAL NOTEBOOK # 9 BUILD AN ATOM: Post-lab questions MAS = 8 ADV = 7 MTS = 6 APP = 4-5 BEG = 0-3 For questions 1-5: You build an atom

More information

Classification of Phase Portraits at Equilibria for u (t) = f( u(t))

Classification of Phase Portraits at Equilibria for u (t) = f( u(t)) Classification of Phase Portraits at Equilibria for u t = f ut Transfer of Local Linearized Phase Portrait Transfer of Local Linearized Stability How to Classify Linear Equilibria Justification of the

More information

Intermediate Differential Equations. Stability and Bifurcation II. John A. Burns

Intermediate Differential Equations. Stability and Bifurcation II. John A. Burns Intermediate Differential Equations Stability and Bifurcation II John A. Burns Center for Optimal Design And Control Interdisciplinary Center for Applied Mathematics Virginia Polytechnic Institute and

More information

Introduction to Partial Differential Equations

Introduction to Partial Differential Equations Introduction to Partial Differential Equations Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) Key Concepts Current Semester 1 / 25 Introduction The purpose of this section is to define

More information

Question: Total. Points:

Question: Total. Points: MATH 308 May 23, 2011 Final Exam Name: ID: Question: 1 2 3 4 5 6 7 8 9 Total Points: 0 20 20 20 20 20 20 20 20 160 Score: There are 9 problems on 9 pages in this exam (not counting the cover sheet). Make

More information

A matrix times a column vector is the linear combination of the columns of the matrix weighted by the entries in the column vector.

A matrix times a column vector is the linear combination of the columns of the matrix weighted by the entries in the column vector. 18.03 Class 33, Apr 28, 2010 Eigenvalues and eigenvectors 1. Linear algebra 2. Ray solutions 3. Eigenvalues 4. Eigenvectors 5. Initial values [1] Prologue on Linear Algebra. Recall [a b ; c d] [x ; y]

More information

Homework Solutions: , plus Substitutions

Homework Solutions: , plus Substitutions Homework Solutions: 2.-2.2, plus Substitutions Section 2. I have not included any drawings/direction fields. We can see them using Maple or by hand, so we ll be focusing on getting the analytic solutions

More information

Lesson 3: Solving Equations A Balancing Act

Lesson 3: Solving Equations A Balancing Act Opening Exercise Let s look back at the puzzle in Lesson 1 with the t-shape and the 100-chart. Jennie came up with a sum of 380 and through the lesson we found that the expression to represent the sum

More information

Department of Mathematics IIT Guwahati

Department 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 information

Math 10C - Fall Final Exam

Math 10C - Fall Final Exam Math 1C - Fall 217 - Final Exam Problem 1. Consider the function f(x, y) = 1 x 2 (y 1) 2. (i) Draw the level curve through the point P (1, 2). Find the gradient of f at the point P and draw the gradient

More information

Chapter #4 EEE8086-EEE8115. Robust and Adaptive Control Systems

Chapter #4 EEE8086-EEE8115. Robust and Adaptive Control Systems Chapter #4 Robust and Adaptive Control Systems Nonlinear Dynamics.... Linear Combination.... Equilibrium points... 3 3. Linearisation... 5 4. Limit cycles... 3 5. Bifurcations... 4 6. Stability... 6 7.

More information

Solving Equations by Factoring. Solve the quadratic equation x 2 16 by factoring. We write the equation in standard form: x

Solving Equations by Factoring. Solve the quadratic equation x 2 16 by factoring. We write the equation in standard form: x 11.1 E x a m p l e 1 714SECTION 11.1 OBJECTIVES 1. Solve quadratic equations by using the square root method 2. Solve quadratic equations by completing the square Here, we factor the quadratic member of

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

Autonomous Systems and Stability

Autonomous Systems and Stability LECTURE 8 Autonomous Systems and Stability An autonomous system is a system of ordinary differential equations of the form 1 1 ( 1 ) 2 2 ( 1 ). ( 1 ) or, in vector notation, x 0 F (x) That is to say, an

More information

Math Section 4.3 Unit Circle Trigonometry

Math Section 4.3 Unit Circle Trigonometry Math 10 - Section 4. Unit Circle Trigonometry An angle is in standard position if its vertex is at the origin and its initial side is along the positive x axis. Positive angles are measured counterclockwise

More information

MATH 251 Examination II April 7, 2014 FORM A. Name: Student Number: Section:

MATH 251 Examination II April 7, 2014 FORM A. Name: Student Number: Section: MATH 251 Examination II April 7, 2014 FORM A Name: Student Number: Section: This exam has 12 questions for a total of 100 points. In order to obtain full credit for partial credit problems, all work must

More information

7 Planar systems of linear ODE

7 Planar systems of linear ODE 7 Planar systems of linear ODE Here I restrict my attention to a very special class of autonomous ODE: linear ODE with constant coefficients This is arguably the only class of ODE for which explicit solution

More information

MATH 251 Examination II July 28, Name: Student Number: Section:

MATH 251 Examination II July 28, Name: Student Number: Section: MATH 251 Examination II July 28, 2008 Name: Student Number: Section: This exam has 9 questions for a total of 100 points. In order to obtain full credit for partial credit problems, all work must be shown.

More information

Chapter 8 Equilibria in Nonlinear Systems

Chapter 8 Equilibria in Nonlinear Systems Chapter 8 Equilibria in Nonlinear Sstems Recall linearization for Nonlinear dnamical sstems in R n : X 0 = F (X) : if X 0 is an equilibrium, i.e., F (X 0 ) = 0; then its linearization is U 0 = AU; A =

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

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

Lesson 6b Rational Exponents & Radical Functions

Lesson 6b Rational Exponents & Radical Functions Lesson 6b Rational Exponents & Radical Functions In this lesson, we will continue our review of Properties of Exponents and will learn some new properties including those dealing with Rational and Radical

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

Leon, Chap. 1. Chap. 2

Leon, Chap. 1. Chap. 2 Linear Algebra for Dierential Equations Math 309.501, 309.502 Spring, 2014 Section 501: MWF 10:2011:10 (BLOC 164) Section 502: MWF 11:3012:20 (BLOC 164) Instructor: Prof. Thomas Vogel Oce: 629C Blocker

More information

Math 266: Phase Plane Portrait

Math 266: Phase Plane Portrait Math 266: Phase Plane Portrait Long Jin Purdue, Spring 2018 Review: Phase line for an autonomous equation For a single autonomous equation y = f (y) we used a phase line to illustrate the equilibrium solutions

More information

Chapter 1: Introduction

Chapter 1: Introduction Chapter 1: Introduction Definition: A differential equation is an equation involving the derivative of a function. If the function depends on a single variable, then only ordinary derivatives appear and

More information

Math 273 (51) - Final

Math 273 (51) - Final Name: Id #: Math 273 (5) - Final Autumn Quarter 26 Thursday, December 8, 26-6: to 8: Instructions: Prob. Points Score possible 25 2 25 3 25 TOTAL 75 Read each problem carefully. Write legibly. Show all

More information

Linear Systems. Class 27. c 2008 Ron Buckmire. TITLE Projection Matrices and Orthogonal Diagonalization CURRENT READING Poole 5.4

Linear Systems. Class 27. c 2008 Ron Buckmire. TITLE Projection Matrices and Orthogonal Diagonalization CURRENT READING Poole 5.4 Linear Systems Math Spring 8 c 8 Ron Buckmire Fowler 9 MWF 9: am - :5 am http://faculty.oxy.edu/ron/math//8/ Class 7 TITLE Projection Matrices and Orthogonal Diagonalization CURRENT READING Poole 5. Summary

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

*****DO NOT WRITE ON THIS PAPER***** Answer these questions in your SPIRAL NOTEBOOK. BUILD AN ATOM: Post-lab questions

*****DO NOT WRITE ON THIS PAPER***** Answer these questions in your SPIRAL NOTEBOOK. BUILD AN ATOM: Post-lab questions *****DO NOT WRITE ON THIS PAPER***** Answer these questions in your SPIRAL NOTEBOOK BUILD AN ATOM: Post-lab questions For questions 1-5: You build an atom that has the following components: 3 protons P

More information

Invariant Manifolds of Dynamical Systems and an application to Space Exploration

Invariant Manifolds of Dynamical Systems and an application to Space Exploration Invariant Manifolds of Dynamical Systems and an application to Space Exploration Mateo Wirth January 13, 2014 1 Abstract In this paper we go over the basics of stable and unstable manifolds associated

More information

Only this exam and a pen or pencil should be on your desk.

Only this exam and a pen or pencil should be on your desk. Lin. Alg. & Diff. Eq., Spring 16 Student ID Circle your section: 31 MWF 8-9A 11 LATIMER LIANG 33 MWF 9-1A 11 LATIMER SHAPIRO 36 MWF 1-11A 37 CORY SHAPIRO 37 MWF 11-1P 736 EVANS WORMLEIGHTON 39 MWF -5P

More information

Applied Calculus. Review Problems for the Final Exam

Applied Calculus. Review Problems for the Final Exam Math135 Study Guide 1 Math 131/135/194, Fall 2004 Applied Calculus Daniel Kaplan, Macalester College Review Problems for the Final Exam Problem 1../DE/102b.tex Problem 3../DE/107.tex Consider the pair

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

MATH 251 Final Examination August 10, 2011 FORM A. Name: Student Number: Section:

MATH 251 Final Examination August 10, 2011 FORM A. Name: Student Number: Section: MATH 251 Final Examination August 10, 2011 FORM A Name: Student Number: Section: This exam has 10 questions for a total of 150 points. In order to obtain full credit for partial credit problems, all work

More information

DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS. MATH 233 SOME SOLUTIONS TO EXAM 2 Fall 2018

DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS. MATH 233 SOME SOLUTIONS TO EXAM 2 Fall 2018 DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS MATH 233 SOME SOLUTIONS TO EXAM 2 Fall 208 Version A refers to the regular exam and Version B to the make-up. Version A. A particle

More information

4 Second-Order Systems

4 Second-Order Systems 4 Second-Order Systems Second-order autonomous systems occupy an important place in the study of nonlinear systems because solution trajectories can be represented in the plane. This allows for easy visualization

More information

Math 215/255 Final Exam (Dec 2005)

Math 215/255 Final Exam (Dec 2005) Exam (Dec 2005) Last Student #: First name: Signature: Circle your section #: Burggraf=0, Peterson=02, Khadra=03, Burghelea=04, Li=05 I have read and understood the instructions below: Please sign: Instructions:.

More information

Introduction to systems of equations

Introduction to systems of equations Introduction to systems of equations A system of equations is a collection of two or more equations that contains the same variables. This is a system of two equations with two variables: In solving a

More information

MCE693/793: Analysis and Control of Nonlinear Systems

MCE693/793: Analysis and Control of Nonlinear Systems MCE693/793: Analysis and Control of Nonlinear Systems Systems of Differential Equations Phase Plane Analysis Hanz Richter Mechanical Engineering Department Cleveland State University Systems of Nonlinear

More information

Slope Fields: Graphing Solutions Without the Solutions

Slope Fields: Graphing Solutions Without the Solutions 8 Slope Fields: Graphing Solutions Without the Solutions Up to now, our efforts have been directed mainly towards finding formulas or equations describing solutions to given differential equations. Then,

More information

Math 216 First Midterm 19 October, 2017

Math 216 First Midterm 19 October, 2017 Math 6 First Midterm 9 October, 7 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

Math 409/509 (Spring 2011)

Math 409/509 (Spring 2011) Math 409/509 (Spring 2011) Instructor: Emre Mengi Study Guide for Homework 2 This homework concerns the root-finding problem and line-search algorithms for unconstrained optimization. Please don t hesitate

More information

MATH 215/255 Solutions to Additional Practice Problems April dy dt

MATH 215/255 Solutions to Additional Practice Problems April dy dt . For the nonlinear system MATH 5/55 Solutions to Additional Practice Problems April 08 dx dt = x( x y, dy dt = y(.5 y x, x 0, y 0, (a Show that if x(0 > 0 and y(0 = 0, then the solution (x(t, y(t of the

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

Announcements Monday, November 13

Announcements Monday, November 13 Announcements Monday, November 13 The third midterm is on this Friday, November 17. The exam covers 3.1, 3.2, 5.1, 5.2, 5.3, and 5.5. About half the problems will be conceptual, and the other half computational.

More information

1. Open polymath: 2. Go to Help, Contents F1 or Press F1

1. Open polymath: 2. Go to Help, Contents F1 or Press F1 Polymath Tutorial Process Fluid Transport 1. Open polymath: 2. Go to Help, Contents F1 or Press F1 1 3. Read the section titled Introduction to Polymath both getting started and Variables and expressions

More information

MA 527 first midterm review problems Hopefully final version as of October 2nd

MA 527 first midterm review problems Hopefully final version as of October 2nd MA 57 first midterm review problems Hopefully final version as of October nd The first midterm will be on Wednesday, October 4th, from 8 to 9 pm, in MTHW 0. It will cover all the material from the classes

More information

1 Ordinary points and singular points

1 Ordinary points and singular points Math 70 honors, Fall, 008 Notes 8 More on series solutions, and an introduction to \orthogonal polynomials" Homework at end Revised, /4. Some changes and additions starting on page 7. Ordinary points and

More information

MATH 251 Final Examination August 14, 2015 FORM A. Name: Student Number: Section:

MATH 251 Final Examination August 14, 2015 FORM A. Name: Student Number: Section: MATH 251 Final Examination August 14, 2015 FORM A Name: Student Number: Section: This exam has 11 questions for a total of 150 points. Show all your work! In order to obtain full credit for partial credit

More information

Sample Questions, Exam 1 Math 244 Spring 2007

Sample Questions, Exam 1 Math 244 Spring 2007 Sample Questions, Exam Math 244 Spring 2007 Remember, on the exam you may use a calculator, but NOT one that can perform symbolic manipulation (remembering derivative and integral formulas are a part of

More information

MATH 294???? FINAL # 4 294UFQ4.tex Find the general solution y(x) of each of the following ODE's a) y 0 = cosecy

MATH 294???? FINAL # 4 294UFQ4.tex Find the general solution y(x) of each of the following ODE's a) y 0 = cosecy 3.1. 1 ST ORDER ODES 1 3.1 1 st Order ODEs MATH 294???? FINAL # 4 294UFQ4.tex 3.1.1 Find the general solution y(x) of each of the following ODE's a) y 0 = cosecy MATH 294 FALL 1990 PRELIM 2 # 4 294FA90P2Q4.tex

More information

Chapter Homogeneous System with Constant Coeffici

Chapter 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 information

Math 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 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 information

Name: ID.NO: Fall 97. PLEASE, BE NEAT AND SHOW ALL YOUR WORK; CIRCLE YOUR ANSWER. NO NOTES, BOOKS, CALCULATORS, TAPE PLAYERS, or COMPUTERS.

Name: ID.NO: Fall 97. PLEASE, BE NEAT AND SHOW ALL YOUR WORK; CIRCLE YOUR ANSWER. NO NOTES, BOOKS, CALCULATORS, TAPE PLAYERS, or COMPUTERS. MATH 303-2/6/97 FINAL EXAM - Alternate WILKERSON SECTION Fall 97 Name: ID.NO: PLEASE, BE NEAT AND SHOW ALL YOUR WORK; CIRCLE YOUR ANSWER. NO NOTES, BOOKS, CALCULATORS, TAPE PLAYERS, or COMPUTERS. Problem

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