Nonlinear Systems of Ordinary Differential Equations
|
|
- Arline Snow
- 7 years ago
- Views:
Transcription
1 Differential Equations Massoud Malek Nonlinear Systems of Ordinary Differential Equations Dynamical System. A dynamical system has a state determined by a collection of real numbers, or more generally by a set of points in an appropriate state space. Small changes in the state of the system correspond to small changes in the numbers. The evolution rule of the dynamical system is a fied rule that describes what future states follow from the current state. The rule is deterministic: for a given time interval only one future state follows from the current state. The mathematical models used to describe the swinging of a clock pendulum, the flow of water in a pipe, or the number of fish each spring in a lake are eamples of dynamical systems. Autonomous System. An autonomous differential equation is a system of ordinary differential equations which does not depend on the independent variable. It is of the form d X(t) = F (X(t)), dt where X takes values in n-dimensional Euclidean space and t is usually time. It is distinguished from systems of differential equations of the form d X(t) = G(X(t), t), dt in which the law governing the rate of motion of a particle depends not only on the particle s location, but also on time; such systems are not autonomous. Autonomous systems are closely related to dynamical systems. Any autonomous system can be transformed into a dynamical system and, using very weak assumptions, a dynamical system can be transformed into an autonomous systems. Jacobian Matri. Consider the function F : IR n IR m, where F ( 1, 2,..., n ) = f 1 ( 1, 2,..., n ) f 2 ( 1, 2,..., n ) f m ( 1, 2,..., n ) The partial derivatives of f 1 ( 1,..., n ),..., f m ( 1,..., n ) (if they eist) can be organized in an m n matri. The Jacobian matri of F ( 1, 2,..., n ) denoted by J F f 1 1 J F ( 1,..., n ) =..... f m 1. is as follows: Its importance lies in the fact that it represents the best linear approimation to a differentiable function near a given point. f 1 n f m n.
2 Massoud Malek Nonlinear Systems of Ordinary Differential Equations Page 2 Qualitative Analysis. Very often it is almost impossible to find eplicitly or implicitly the solutions of a system (specially nonlinear ones). The qualitative approach as well as numerical one are important since they allow us to make conclusions regardless whether we know or not the solutions. Nullclines and Equilibrium Points Consider the system of first order ordinary differential equations: 1 = f 1 ( 1, 2..., n ) 2 = f 2 ( 1, 2..., n )... n = f n ( 1, 2..., n ). The j -nullcline is the set of points which satisfy f j ( 1, 2,..., n ) = 0. The intersection point of all the nullclines is called an equilibrium point or fied point of the system. The Jacobian matri with constant entries, is identified with the matri of a linear systems. Near a fied point ( 1, 2,..., n), the dynamics of the nonlinear system are qualitatively similar to the dynamics of the linear system associated with the Jacobian matri J( 1, 2,..., n), provided its eigenvalues λ js have nonzero real parts. Fied points with a Jacobian matri such that Re(λ j ) 0 are called hyperbolic fied points. Otherwise, they are non-hyperbolic fied points, whose stabilities must be determined directly. Eample 1. Consider the system: (t) = (1 ) y, y (t) = 2y(1 y2 2 ) 32 y. The -nullclines are given by (t) = (1 ) y = 0 which is equivalent to = 0 or y = + 1. The y-nullclines are given by y (t) = 2y(1 y 2 ) 3y = 0 which is equivalent to y = 0 or y 2 = 2. Eample 2. Consider the model describing two competing species: (t) = (1 y), y (t) = 2y (1 y2 32 ). The -nullclines are = 0 or y = + 1. The y-nullclines are y = 0 or y = The equilibrium points are (0, 0), (0, 2), (1, 0), and ( 1 2, 1 2 ). The components of the velocity vectors are (t) and y (t). These vectors give the direction of the motion along the trajectories. We have the four natural directions (left, right, up, and down) and the other four directions (left-down, left-up, right-down, and right-up). These directions are obtained by looking at the signs of (t) and y (t) and whether they are equal to 0. If both are zero, then we have an equilibrium point. Note that along the -nullcline the velocity vectors are vertical while along the y-nullcline the velocity vectors are horizontal. Note that as long as we are traveling along a nullcline without crossing an equilibrium point, then the direction of the velocity vector must be the same. Once we cross an equilibrium point, then we may have a change in the direction (from up to down, or right to left, and vice-versa).
3 Massoud Malek Nonlinear Systems of Ordinary Differential Equations Page 3 Nullclines - Fied Points - Velocity Vectors Eample 1. Eample 2. In order to find the direction of the velocity vectors along the nullclines, we pick a point on the nullcline and find the direction of the velocity vector at that point. The velocity vector along the segment of the nullcline delimited by equilibrium points which contains the given point will have the same direction. For eample, consider the point (1/3, 1) on the y-nullcline y = in the second eample. The velocity vector at this point is ( 1/9, 0). Therefore the velocity vector at any point on the line y = 3 + 2, with > 1/3, is horizontal and points to the left (since = 1/9 < 0). The picture below gives the nullclines and the velocity vectors along them. Velocity Vectors Remark. The point (0, 0) is a fied point of any linear system of ordinary differential equation, but a nonlinear system may have neither fied points nor nullclines. Eample 3. (t) = 2 + 1, y (t) = (y 1). No -nullcline. The y-nullcline are = 0 or y 1. No fied point. Eample 4. (t) = 2 + y 2 1, y (t) = y + 2. The -nullcline is the unit circle. The y-nullcline is the line y = + 2. The nullclines do not intersect.
4 Massoud Malek Nonlinear Systems of Ordinary Differential Equations Page 4 Nonlinear Autonomous Systems of Two Equations Most of the interesting differential equations are non-linear and, with a few eceptions, cannot be solved eactly. Approimate solutions are arrived at using computer approimations. A first order nonlinear autonomous system is: (t) = F (, y), At the site: y (t) = G(, y). they use Java to show you graphs of solutions of first order nonlinear autonomous systems of two equations. To see the graphs of the vector field and flow curves go to Here are a few eamples of second order nonlinear autonomous systems: Equation of motion of point mass in the (,y)-plane under gravitational force: tt = kr 3, y tt = kyr 3, where r = 2 + y 2. Equation of motion of a point mass in the (,y)-plane under central force: Equations of motion of a projectile: Linearization Technique tt = f(r), y tt = yf(r), where r = 2 + y 2. tt = f(y)g(v) t, y tt = f(y)g(v)y t a, where v = ( ) 2 + (y ) 2. Consider the autonomous nonlinear system (t) = F (, y), y (t) = G(, y). with (, y ) a fied point. We would like to find the closest linear system when (, y) is close to (, y ). In order to do that we need to approimate the functions F (, y) and G(, y) around the equilibrium point (, y ) by its tangent around that fied point. From multi-variable calculus, we know that when (, y) is close to (, y ), the nonlinear system may be approimated by the system d d t (t) = F (, y) F (, y ) + (, y )( ) + y (, y )(y y ) d d t y(t) = G(, y) G(, y ) + G (, y )( ) + G y (, y )(y y ). Since (, y ) is a fied point, we have F (, y ) = G(, y ) = 0. Thus d d t (t) (, y )( ) + y (, y )(y y ) d G y(t) d t (, y )( ) + G y (, y )(y y ).
5 Massoud Malek Nonlinear Systems of Ordinary Differential Equations Page 5 This is a linear system. Its coefficient matri is J = (, y ) G (, y ) y (, y ). G y (, y ) This matri is just the Jacobian matri of the system at the fied point (, y ). Thus y = (, y ) G (, y ) y (, y ). G y (, y ) y y Note. If the equilibrium point (, y ) (0, 0), then by choosing u = and v = y y, we may the system to a new system with (0, 0) as a fied point. Topological Classification Linear stability analysis works for a hyperbolic fied points. The nonlinear system s phase portrait near the fied point is topologically unchanged due to small perturbations, and its dynamics are structurally stable or robust. Poincare-Lyapunov Theorem. If the eigenvalues of the Jacobian matri evaluated at the fied point are not equal zero or are not pure imaginary numbers, then the trajectories of the system around the equilibrium point behave the same way as the trajectories of the associated linear system. 1. If the eigenvalues are negative or comple with negative real part, then the fied point is a sink (that is all the solutions will die at the equilibrium point). Note that if the eigenvalues are comple, then the solutions will spiral around the equilibrium point. 2. If the eigenvalues are positive or comple with positive real part, then the fied point is a source (this means that the solutions on the trajectories will move away from the equilibrium point). Note that if the eigenvalues are comple, then the solutions will spiral away from the fied point. 3. If the eigenvalues are real number with different sign (one positive and one negative), then the equilibrium point is a saddle point. In fact, there will be two solutions which approach the equilibrium point as t, and two more solutions which approach the equilibrium point as t. For the linear system theses solutions are lines, but for the nonlinear system they are not in general. These four solutions are called separatri. Let p = trace[j(, y )] and q = det[j(, y )], then hyperbolic fied points are classified as follows: Repellers (Sources) Unstable p > 0, q > 0 Re(λ 1 ) > 0, Re(λ 2 ) > 0 Attractors (Sinks) Stable p < 0, q > 0 Re(λ 1 ) < 0, Re(λ 2 ) < 0 Saddle Points Unstable q < 0 Re(λ 1 ) < 0, Re(λ 2 ) > 0 Linear stability analysis may fail for a non-hyperbolic fied point: Re(λ 1 ) = 0 and Re(λ 2 ) = 0 or at least one eigenvalue is zero. The classifications for the fied points of a nonlinear system are summarized in the
6 Massoud Malek Nonlinear Systems of Ordinary Differential Equations Page 6 following diagram: p Saddle points Saddle points Repellers (Sources) Attractors (Sinks) q Eample 5. Consider the nonlinear system (t) = F (, y) = 3, y (t) = G(, y) = 2y. The solution is: d 3 = dt (t) = ± (1 C 1 e 2t ) 1 ; dy y = 2 dt y(t) = C 2 e 2t. The fied points are the intersections of the nullclines y = 0 (the -ais) with = 1, = 0, and = 1. Stability at Fied Points [ ] 3 The Jacobian matri is J(, y) = with 0 2 [ ] [ ] [ ] J( 1, 0) =, J(0, 0) =, and 2 0 J(1, 0) = Note that around the fied points ( 1, 0), (0, 0), and (1, 0), the nonlinear system should behave like the linear systems: [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] y =, 0 2 y y =, and 0 2 y y = 0 2 y respectively. Since all the eigenvalues have nonzero real part, we conclude that all three fied points are hyperbolic. Consequently, the nonlinear system has a stable node (attractor) at (0, 0) and saddle points at ( 1, 0) and (1, 0).
7 Massoud Malek Nonlinear Systems of Ordinary Differential Equations Page 7 P hase P ortrait Eample 6. Consider the following second order nonlinear equation known as Van der Pol equation d 2 dt 2 (1 2 ) d dt + = 0. This can be translated into the following system. Set y = d. Then we have dt (t) = y, y (t) = + (1 2 )y. The -nullcline is given by d = y = 0. Hence the -nullcline is the -ais. dt The y-nullcline is given by dy Thus (, y ) = (0, 0) is the only fied point. The Jacobian matri is dt = + (1 2 )y. Hence the y-nullcline is the curve y = [ ] 0 1 J(, y) = 1 2y 1 2 with J(0, 0) = [ ] Net picture shows the graphs of the solutions (t) and y(t) for the initial value (0, 4). The linear system close to the original nonlinear system around the fied point (0, 0) is (t) = y, y (t) = + y. or ( ) = y [ ] ( ). y The eigenvalues of this system are 1 ± 3 i. Since the real part is positive, the solutions 2 of the linear system spiral away from the origin. Eample 7. Finally, consider the following problem: = y y = 9 sin y 5
8 Massoud Malek Nonlinear Systems of Ordinary Differential Equations Page 8 [ ] 0 1 with J = 9 cos 1 5 -nullcline: y = 0 y-nullclines: y = 45 sin fied points: (nπ, 0) n =, 3, 2,0, 1, 2, 3, Nullclines and F ied P oints P hase P ortrait
Eigenvalues, Eigenvectors, and Differential Equations
Eigenvalues, Eigenvectors, and Differential Equations William Cherry April 009 (with a typo correction in November 05) The concepts of eigenvalue and eigenvector occur throughout advanced mathematics They
More informationOrbits of the Lennard-Jones Potential
Orbits of the Lennard-Jones Potential Prashanth S. Venkataram July 28, 2012 1 Introduction The Lennard-Jones potential describes weak interactions between neutral atoms and molecules. Unlike the potentials
More informationChapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs
Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s
More informationStudent name: Earlham College. Fall 2011 December 15, 2011
Student name: Earlham College MATH 320: Differential Equations Final exam - In class part Fall 2011 December 15, 2011 Instructions: This is a regular closed-book test, and is to be taken without the use
More informationAP Calculus AB 2004 Scoring Guidelines
AP Calculus AB 4 Scoring Guidelines The materials included in these files are intended for noncommercial use by AP teachers for course and eam preparation; permission for any other use must be sought from
More information3.2 Sources, Sinks, Saddles, and Spirals
3.2. Sources, Sinks, Saddles, and Spirals 6 3.2 Sources, Sinks, Saddles, and Spirals The pictures in this section show solutions to Ay 00 C By 0 C Cy D 0. These are linear equations with constant coefficients
More information2008 AP Calculus AB Multiple Choice Exam
008 AP Multiple Choice Eam Name 008 AP Calculus AB Multiple Choice Eam Section No Calculator Active AP Calculus 008 Multiple Choice 008 AP Calculus AB Multiple Choice Eam Section Calculator Active AP Calculus
More informationMATH 10550, EXAM 2 SOLUTIONS. x 2 + 2xy y 2 + x = 2
MATH 10550, EXAM SOLUTIONS (1) Find an equation for the tangent line to at the point (1, ). + y y + = Solution: The equation of a line requires a point and a slope. The problem gives us the point so we
More information4 Lyapunov Stability Theory
4 Lyapunov Stability Theory In this section we review the tools of Lyapunov stability theory. These tools will be used in the next section to analyze the stability properties of a robot controller. We
More informationA First Course in Elementary Differential Equations. Marcel B. Finan Arkansas Tech University c All Rights Reserved
A First Course in Elementary Differential Equations Marcel B. Finan Arkansas Tech University c All Rights Reserved 1 Contents 1 Basic Terminology 4 2 Qualitative Analysis: Direction Field of y = f(t, y)
More informationPhase Portraits for u = A u
Phase Portraits for u = A u How to Construct a Phase Portrait Badly Threaded Solution Curves Solution Curve Tangent Matching Phase Portrait Illustration Phase plot by computer Revised Computer Phase plot
More informationAP Calculus AB First Semester Final Exam Practice Test Content covers chapters 1-3 Name: Date: Period:
AP Calculus AB First Semester Final Eam Practice Test Content covers chapters 1- Name: Date: Period: This is a big tamale review for the final eam. Of the 69 questions on this review, questions will be
More informationLecture L22-2D Rigid Body Dynamics: Work and Energy
J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L - D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L-3 for
More informationDynamical Systems Analysis II: Evaluating Stability, Eigenvalues
Dynamical Systems Analysis II: Evaluating Stability, Eigenvalues By Peter Woolf pwoolf@umich.edu) University of Michigan Michigan Chemical Process Dynamics and Controls Open Textbook version 1.0 Creative
More information3. Reaction Diffusion Equations Consider the following ODE model for population growth
3. Reaction Diffusion Equations Consider the following ODE model for population growth u t a u t u t, u 0 u 0 where u t denotes the population size at time t, and a u plays the role of the population dependent
More informationMath 2280 - Assignment 6
Math 2280 - Assignment 6 Dylan Zwick Spring 2014 Section 3.8-1, 3, 5, 8, 13 Section 4.1-1, 2, 13, 15, 22 Section 4.2-1, 10, 19, 28 1 Section 3.8 - Endpoint Problems and Eigenvalues 3.8.1 For the eigenvalue
More informationAP Calculus BC 2008 Scoring Guidelines
AP Calculus BC 8 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to college
More informationNumerical Solution of Differential Equations
Numerical Solution of Differential Equations Dr. Alvaro Islas Applications of Calculus I Spring 2008 We live in a world in constant change We live in a world in constant change We live in a world in constant
More informationDifferentiation of vectors
Chapter 4 Differentiation of vectors 4.1 Vector-valued functions In the previous chapters we have considered real functions of several (usually two) variables f : D R, where D is a subset of R n, where
More informationPower functions: f(x) = x n, n is a natural number The graphs of some power functions are given below. n- even n- odd
5.1 Polynomial Functions A polynomial unctions is a unction o the orm = a n n + a n-1 n-1 + + a 1 + a 0 Eample: = 3 3 + 5 - The domain o a polynomial unction is the set o all real numbers. The -intercepts
More informationBy Clicking on the Worksheet you are in an active Math Region. In order to insert a text region either go to INSERT -TEXT REGION or simply
Introduction and Basics Tet Regions By Clicking on the Worksheet you are in an active Math Region In order to insert a tet region either go to INSERT -TEXT REGION or simply start typing --the first time
More informationAP Calculus AB 2007 Scoring Guidelines Form B
AP Calculus AB 7 Scoring Guidelines Form B The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to
More informationMark Howell Gonzaga High School, Washington, D.C.
Be Prepared for the Calculus Eam Mark Howell Gonzaga High School, Washington, D.C. Martha Montgomery Fremont City Schools, Fremont, Ohio Practice eam contributors: Benita Albert Oak Ridge High School,
More informationCh 7 Kinetic Energy and Work. Question: 7 Problems: 3, 7, 11, 17, 23, 27, 35, 37, 41, 43
Ch 7 Kinetic Energy and Work Question: 7 Problems: 3, 7, 11, 17, 23, 27, 35, 37, 41, 43 Technical definition of energy a scalar quantity that is associated with that state of one or more objects The state
More informationMatLab - Systems of Differential Equations
Fall 2015 Math 337 MatLab - Systems of Differential Equations This section examines systems of differential equations. It goes through the key steps of solving systems of differential equations through
More informationWorksheet 1. What You Need to Know About Motion Along the x-axis (Part 1)
Worksheet 1. What You Need to Know About Motion Along the x-axis (Part 1) In discussing motion, there are three closely related concepts that you need to keep straight. These are: If x(t) represents the
More informationUnderstanding Poles and Zeros
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.14 Analysis and Design of Feedback Control Systems Understanding Poles and Zeros 1 System Poles and Zeros The transfer function
More informationLIMITS AND CONTINUITY
LIMITS AND CONTINUITY 1 The concept of it Eample 11 Let f() = 2 4 Eamine the behavior of f() as approaches 2 2 Solution Let us compute some values of f() for close to 2, as in the tables below We see from
More informationStudent Performance Q&A:
Student Performance Q&A: 2008 AP Calculus AB and Calculus BC Free-Response Questions The following comments on the 2008 free-response questions for AP Calculus AB and Calculus BC were written by the Chief
More informationAP Calculus BC 2001 Free-Response Questions
AP Calculus BC 001 Free-Response Questions The materials included in these files are intended for use by AP teachers for course and exam preparation in the classroom; permission for any other use must
More informationCalculus 1st Semester Final Review
Calculus st Semester Final Review Use the graph to find lim f ( ) (if it eists) 0 9 Determine the value of c so that f() is continuous on the entire real line if f ( ) R S T, c /, > 0 Find the limit: lim
More informationAutonomous Equations / Stability of Equilibrium Solutions. y = f (y).
Autonomous Equations / Stabilit of Equilibrium Solutions First order autonomous equations, Equilibrium solutions, Stabilit, Longterm behavior of solutions, direction fields, Population dnamics and logistic
More informationAbout the Gamma Function
About the Gamma Function Notes for Honors Calculus II, Originally Prepared in Spring 995 Basic Facts about the Gamma Function The Gamma function is defined by the improper integral Γ) = The integral is
More informationMark Howell Gonzaga High School, Washington, D.C.
Be Prepared for the Calculus Exam Mark Howell Gonzaga High School, Washington, D.C. Martha Montgomery Fremont City Schools, Fremont, Ohio Practice exam contributors: Benita Albert Oak Ridge High School,
More informationThe aerodynamic center
The aerodynamic center In this chapter, we re going to focus on the aerodynamic center, and its effect on the moment coefficient C m. 1 Force and moment coefficients 1.1 Aerodynamic forces Let s investigate
More informationMechanics 1: Conservation of Energy and Momentum
Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation
More informationExample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum. asin. k, a, and b. We study stability of the origin x
Lecture 4. LaSalle s Invariance Principle We begin with a motivating eample. Eample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum Dynamics of a pendulum with friction can be written
More informationExponential and Logarithmic Functions
Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,
More informationSolutions to old Exam 1 problems
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
More informationChapter 4 One Dimensional Kinematics
Chapter 4 One Dimensional Kinematics 41 Introduction 1 4 Position, Time Interval, Displacement 41 Position 4 Time Interval 43 Displacement 43 Velocity 3 431 Average Velocity 3 433 Instantaneous Velocity
More informationObjectives. Materials
Activity 4 Objectives Understand what a slope field represents in terms of Create a slope field for a given differential equation Materials TI-84 Plus / TI-83 Plus Graph paper Introduction One of the ways
More information0 0 such that f x L whenever x a
Epsilon-Delta Definition of the Limit Few statements in elementary mathematics appear as cryptic as the one defining the limit of a function f() at the point = a, 0 0 such that f L whenever a Translation:
More informationSecond Order Linear Partial Differential Equations. Part I
Second Order Linear Partial Differential Equations Part I Second linear partial differential equations; Separation of Variables; - point boundary value problems; Eigenvalues and Eigenfunctions Introduction
More informationChapter 28 Fluid Dynamics
Chapter 28 Fluid Dynamics 28.1 Ideal Fluids... 1 28.2 Velocity Vector Field... 1 28.3 Mass Continuity Equation... 3 28.4 Bernoulli s Principle... 4 28.5 Worked Examples: Bernoulli s Equation... 7 Example
More informationGround Rules. PC1221 Fundamentals of Physics I. Kinematics. Position. Lectures 3 and 4 Motion in One Dimension. Dr Tay Seng Chuan
Ground Rules PC11 Fundamentals of Physics I Lectures 3 and 4 Motion in One Dimension Dr Tay Seng Chuan 1 Switch off your handphone and pager Switch off your laptop computer and keep it No talking while
More informationAn Introduction to Calculus. Jackie Nicholas
Mathematics Learning Centre An Introduction to Calculus Jackie Nicholas c 2004 University of Sydney Mathematics Learning Centre, University of Sydney 1 Some rules of differentiation and how to use them
More informationAP Calculus AB 2005 Scoring Guidelines Form B
AP Calculus AB 5 coring Guidelines Form B The College Board: Connecting tudents to College uccess The College Board is a not-for-profit membership association whose mission is to connect students to college
More informationImplicit Differentiation
Revision Notes 2 Calculus 1270 Fall 2007 INSTRUCTOR: Peter Roper OFFICE: LCB 313 [EMAIL: roper@math.utah.edu] Standard Disclaimer These notes are not a complete review of the course thus far, and some
More informationCalculus AB 2014 Scoring Guidelines
P Calculus B 014 Scoring Guidelines 014 The College Board. College Board, dvanced Placement Program, P, P Central, and the acorn logo are registered trademarks of the College Board. P Central is the official
More informationThe integrating factor method (Sect. 2.1).
The integrating factor method (Sect. 2.1). Overview of differential equations. Linear Ordinary Differential Equations. The integrating factor method. Constant coefficients. The Initial Value Problem. Variable
More information19.7. Applications of Differential Equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Applications of Differential Equations 19.7 Introduction Blocks 19.2 to 19.6 have introduced several techniques for solving commonly-occurring firstorder and second-order ordinary differential equations.
More informationMEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145:
MEMORANDUM To: All students taking the CLC Math Placement Eam From: CLC Mathematics Department Subject: What to epect on the Placement Eam Date: April 0 Placement into MTH 45 Solutions This memo is an
More informationAP Calculus BC 2013 Free-Response Questions
AP Calculus BC 013 Free-Response Questions About the College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in
More information88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a
88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small
More informationMathematics 31 Pre-calculus and Limits
Mathematics 31 Pre-calculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals
More informationwww.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates
Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c
More informationSECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA
SECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA This handout presents the second derivative test for a local extrema of a Lagrange multiplier problem. The Section 1 presents a geometric motivation for the
More informationNumerical Methods for Differential Equations
Numerical Methods for Differential Equations Chapter 1: Initial value problems in ODEs Gustaf Söderlind and Carmen Arévalo Numerical Analysis, Lund University Textbooks: A First Course in the Numerical
More informationHomework 2 Solutions
Homework Solutions 1. (a) Find the area of a regular heagon inscribed in a circle of radius 1. Then, find the area of a regular heagon circumscribed about a circle of radius 1. Use these calculations to
More information1.5 SOLUTION SETS OF LINEAR SYSTEMS
1-2 CHAPTER 1 Linear Equations in Linear Algebra 1.5 SOLUTION SETS OF LINEAR SYSTEMS Many of the concepts and computations in linear algebra involve sets of vectors which are visualized geometrically as
More informationRoots of equation fx are the values of x which satisfy the above expression. Also referred to as the zeros of an equation
LECTURE 20 SOLVING FOR ROOTS OF NONLINEAR EQUATIONS Consider the equation f = 0 Roots of equation f are the values of which satisfy the above epression. Also referred to as the zeros of an equation f()
More informationSystem of First Order Differential Equations
CHAPTER System of First Order Differential Equations In this chapter, we will discuss system of first order differential equations. There are many applications that involving find several unknown functions
More information3 Contour integrals and Cauchy s Theorem
3 ontour integrals and auchy s Theorem 3. Line integrals of complex functions Our goal here will be to discuss integration of complex functions = u + iv, with particular regard to analytic functions. Of
More informationPhysics 1120: Simple Harmonic Motion Solutions
Questions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Physics 1120: Simple Harmonic Motion Solutions 1. A 1.75 kg particle moves as function of time as follows: x = 4cos(1.33t+π/5) where distance is measured
More informationScalar Valued Functions of Several Variables; the Gradient Vector
Scalar Valued Functions of Several Variables; the Gradient Vector Scalar Valued Functions vector valued function of n variables: Let us consider a scalar (i.e., numerical, rather than y = φ(x = φ(x 1,
More informationConnecting Transformational Geometry and Transformations of Functions
Connecting Transformational Geometr and Transformations of Functions Introductor Statements and Assumptions Isometries are rigid transformations that preserve distance and angles and therefore shapes.
More informationRoots of Equations (Chapters 5 and 6)
Roots of Equations (Chapters 5 and 6) Problem: given f() = 0, find. In general, f() can be any function. For some forms of f(), analytical solutions are available. However, for other functions, we have
More informationChapter 2. Parameterized Curves in R 3
Chapter 2. Parameterized Curves in R 3 Def. A smooth curve in R 3 is a smooth map σ : (a, b) R 3. For each t (a, b), σ(t) R 3. As t increases from a to b, σ(t) traces out a curve in R 3. In terms of components,
More informationAP Calculus BC 2006 Free-Response Questions
AP Calculus BC 2006 Free-Response Questions The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to
More informationThe dynamic equation for the angular motion of the wheel is R w F t R w F w ]/ J w
Chapter 4 Vehicle Dynamics 4.. Introduction In order to design a controller, a good representative model of the system is needed. A vehicle mathematical model, which is appropriate for both acceleration
More informationy intercept Gradient Facts Lines that have the same gradient are PARALLEL
CORE Summar Notes Linear Graphs and Equations = m + c gradient = increase in increase in intercept Gradient Facts Lines that have the same gradient are PARALLEL If lines are PERPENDICULAR then m m = or
More informationVector Spaces; the Space R n
Vector Spaces; the Space R n Vector Spaces A vector space (over the real numbers) is a set V of mathematical entities, called vectors, U, V, W, etc, in which an addition operation + is defined and in which
More informationAPPLIED MATHEMATICS ADVANCED LEVEL
APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications
More informationApplications of Second-Order Differential Equations
Applications of Second-Order Differential Equations Second-order linear differential equations have a variety of applications in science and engineering. In this section we explore two of them: the vibration
More informationCHAPTER 2. Eigenvalue Problems (EVP s) for ODE s
A SERIES OF CLASS NOTES FOR 005-006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 4 A COLLECTION OF HANDOUTS ON PARTIAL DIFFERENTIAL EQUATIONS
More informationLinear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices
MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two
More informationReview of Fundamental Mathematics
Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools
More informationLet s first see how precession works in quantitative detail. The system is illustrated below: ...
lecture 20 Topics: Precession of tops Nutation Vectors in the body frame The free symmetric top in the body frame Euler s equations The free symmetric top ala Euler s The tennis racket theorem As you know,
More information1 2 3 1 1 2 x = + x 2 + x 4 1 0 1
(d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which
More informationA QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS
A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors
More informationMulti-variable Calculus and Optimization
Multi-variable Calculus and Optimization Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Multi-variable Calculus and Optimization 1 / 51 EC2040 Topic 3 - Multi-variable Calculus
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More informationC B A T 3 T 2 T 1. 1. What is the magnitude of the force T 1? A) 37.5 N B) 75.0 N C) 113 N D) 157 N E) 192 N
Three boxes are connected by massless strings and are resting on a frictionless table. Each box has a mass of 15 kg, and the tension T 1 in the right string is accelerating the boxes to the right at a
More informationExistence of Traveling Wave Solutions
COMM. MATH. SCI. Vol. 4, No. 4, pp. 731 739 c 2006 International Press EXISTENCE OF TRAVELING WAVE SOLUTIONS IN A HYPERBOLIC-ELLIPTIC SYSTEM OF EQUATIONS M. B. A. MANSOUR Abstract. In this paper we discuss
More informationAlgebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123
Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from
More informationSection 1-4 Functions: Graphs and Properties
44 1 FUNCTIONS AND GRAPHS I(r). 2.7r where r represents R & D ependitures. (A) Complete the following table. Round values of I(r) to one decimal place. r (R & D) Net income I(r).66 1.2.7 1..8 1.8.99 2.1
More informationSample Questions for the AP Physics 1 Exam
Sample Questions for the AP Physics 1 Exam Sample Questions for the AP Physics 1 Exam Multiple-choice Questions Note: To simplify calculations, you may use g 5 10 m/s 2 in all problems. Directions: Each
More informationJournal of Engineering Science and Technology Review 2 (1) (2009) 76-81. Lecture Note
Journal of Engineering Science and Technology Review 2 (1) (2009) 76-81 Lecture Note JOURNAL OF Engineering Science and Technology Review www.jestr.org Time of flight and range of the motion of a projectile
More informationWork as the Area Under a Graph of Force vs. Displacement
Work as the Area Under a Graph of vs. Displacement Situation A. Consider a situation where an object of mass, m, is lifted at constant velocity in a uniform gravitational field, g. The applied force is
More informationF = ma. F = G m 1m 2 R 2
Newton s Laws The ideal models of a particle or point mass constrained to move along the x-axis, or the motion of a projectile or satellite, have been studied from Newton s second law (1) F = ma. In the
More informationTOPIC 4: DERIVATIVES
TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the
More informationLecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20
Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding
More informationAP Calculus AB 2004 Free-Response Questions
AP Calculus AB 2004 Free-Response Questions The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use must be
More informationOscillations. Vern Lindberg. June 10, 2010
Oscillations Vern Lindberg June 10, 2010 You have discussed oscillations in Vibs and Waves: we will therefore touch lightly on Chapter 3, mainly trying to refresh your memory and extend the concepts. 1
More information