Student name: Earlham College. Fall 2011 December 15, 2011

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Student name: Earlham College. Fall 2011 December 15, 2011"

Transcription

1 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 of notes, books, or other reference materials. This portion of the test adds up to 30 points each question (or its numbered parts) is worth 3 points. (1) Determine whether the following pairs of functions are linearly dependent or independent on the indicated interval (give reasons): (a) f(t) = t, g(t) = 5 t on the interval 1 < t < 1. (b) f(t) = e t, g(t) = 5e t on the interval 1 < t < 1. (2) Let y(t) denote the solution of the initial value problem y 3y 4y = 0, y(0) = 2, y (0) = α Find the value of α if lim t y(t) = 0 (3) Find the general solution of the system [ 2 1 u = 1 2 (4) Discuss what would be a suitable form of a particular solution when using the method of undetermined coefficients to solve y + 4y = sin(2t) + t. Give reasons. (5) If y 1, y 2 are linearly independent solutions of t 3 y + 2ty + te t y = 0, and if W (y 1, y 2 )(2) = 2e, find lim t W (y 1, y 2 )(t). (6) Give brief answers to each of the following. (a) Determine whether the differential equation y = x3 + 1 is exact. x 2 y x2 (b) Determine whether the differential equation in (a) is linear or nonlinear, homogeneous or nonhomogeneous. (c) Give an example of a 1st order linear system of differential equations whose equilibrium point is not at the origin of the phase space. Justify your claim. (d) A simple model for the interaction between two competing species is given by the following system of differential equations dx dy = 3x 2xy, = 2.5y xy dt dt where x(t), y(t) represent the populations. Find the non-zero equilibrium point and determine the trajectory directions in the phase-plane (first quadrant only). ] u End of test

2 Student name: Earlham College MATH 320 : Differential Equations : Fall 2011 Take home portion of the final exam. Pick up from the Science Library, starting sround noon December 12. Complete and return to Science Library within 24 hours after checkout, or before library closing time, whichever comes first. Test must be taken between December (by noon). Instructions: Answer all questions on separate paper not on this sheet! Show all steps. You may use the following reference materials: The textbook, your own class notes and homework, supplementary handouts given in class, any materials posted on the class website that were prepared for this class, and a calculator. Prohibited materials: Any other reference sources, including electronic, printed, written or verbal. This test adds up to 33 points. It consists of questions numbered (1) to (5). (1) [5 pts. each 2] Find the general solution of each of the following (a) y y y + y = 4 sin t (b) (x 2 + y 2 ) dy = 2x(2x + y) dx (2) [5 pts.] Use the method of reduction of order to find the general solution of 2t 2 y + ty 3y = 0 given that one solution is y 1 = 1/t. (3) [6 pts.] Consider the nonlinear 1st order system: [ ] [ d x (x y)(3 x y) = dt y x(1 y) (a) Find all the equilibrium solutions. (b) Linearize the system around each equilibrium solution. (c) Pick any one equilibrium point and classify its type (e.g., node, center, focus, etc.) and determine the solution behavior in its vicinity (stable, unstable). (4) [6 pts.] A mass of 0.25 kg is attached to a spring and causes it to stretch cm. The mass is pulled down 4 cm from its equilibrium position and given an initial downward velocity of 16 cm/sec. (a) Write an initial value problem that models this situation. (b) Solve the IVP you setup in part (a). (c) Find the time t 0 when the mass first returns to its equilibrium position. ] Page 1 of 2

3 (5) [6 pts.] (a) Consider a second order linear homogeneous differential equation of the form P (x)y + Q(x)y + R(x)y = 0 (i) This equation can be made exact by multiplying by an integrating factor µ(x) such that µ(x)p (x)y + µ(x)q(x)y + µ(x)r(x)y = 0 can be written in the form [µ(x)p (x)y ] + [f(x)y] = 0 By equating coefficients in these two equations and eliminating f(x), show that µ must satisfy P µ + (2P Q)µ + (P Q + R)µ = 0 This equation is known as the adjoint of the original equation. (b) An equation of the form (i) is self-adjoint if its adjoint is the same as the original equation. Show that a necessary condition for (i) to be self-adjoint is that P (x) = Q(x). End of test

4

5

6

7

8

9

10

11

Math 2280 Section 002 [SPRING 2013] 1

Math 2280 Section 002 [SPRING 2013] 1 Math 2280 Section 002 [SPRING 2013] 1 Today well learn about a method for solving systems of differential equations, the method of elimination, that is very similar to the elimination methods we learned

More information

AP Calculus BC 2001 Free-Response Questions

AP 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 information

x + 5 x 2 + x 2 dx Answer: ln x ln x 1 + c

x + 5 x 2 + x 2 dx Answer: ln x ln x 1 + c . Evaluate the given integral (a) 3xe x2 dx 3 2 e x2 + c (b) 3 x ln xdx 2x 3/2 ln x 4 3 x3/2 + c (c) x + 5 x 2 + x 2 dx ln x + 2 + 2 ln x + c (d) x sin (πx) dx x π cos (πx) + sin (πx) + c π2 (e) 3x ( +

More information

Methods of Solution of Selected Differential Equations Carol A. Edwards Chandler-Gilbert Community College

Methods of Solution of Selected Differential Equations Carol A. Edwards Chandler-Gilbert Community College Methods of Solution of Selected Differential Equations Carol A. Edwards Chandler-Gilbert Community College Equations of Order One: Mdx + Ndy = 0 1. Separate variables. 2. M, N homogeneous of same degree:

More information

Nonlinear Systems of Ordinary Differential Equations

Nonlinear Systems of Ordinary Differential Equations 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

More information

First Order Non-Linear Equations

First Order Non-Linear Equations First Order Non-Linear Equations We will briefly consider non-linear equations. In general, these may be much more difficult to solve than linear equations, but in some cases we will still be able to solve

More information

Name: Exam 1. y = 10 t 2. y(0) = 3

Name: Exam 1. y = 10 t 2. y(0) = 3 Instructions. Answer each of the questions on your own paper. Be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without

More information

Homework #2 Solutions

Homework #2 Solutions MAT Spring Problems Section.:, 8,, 4, 8 Section.5:,,, 4,, 6 Extra Problem # Homework # Solutions... Sketch likely solution curves through the given slope field for dy dx = x + y...8. Sketch likely solution

More information

EXAM. Practice Questions for Exam #2. Math 3350, Spring April 3, 2004 ANSWERS

EXAM. Practice Questions for Exam #2. Math 3350, Spring April 3, 2004 ANSWERS EXAM Practice Questions for Exam #2 Math 3350, Spring 2004 April 3, 2004 ANSWERS i Problem 1. Find the general solution. A. D 3 (D 2)(D 3) 2 y = 0. The characteristic polynomial is λ 3 (λ 2)(λ 3) 2. Thus,

More information

Differential Equations Handout A

Differential Equations Handout A Differential Equations Handout A 1. For each of the differential equations below do the following: Sketch the slope field in the t x plane (t on the horizontal axis and x on the vertical) and, on the same

More information

SECOND-ORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS

SECOND-ORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS L SECOND-ORDER LINEAR HOOGENEOUS DIFFERENTIAL EQUATIONS SECOND-ORDER LINEAR HOOGENEOUS DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS A second-order linear differential equation is one of the form d

More information

tegrals as General & Particular Solutions

tegrals as General & Particular Solutions tegrals as General & Particular Solutions dy dx = f(x) General Solution: y(x) = f(x) dx + C Particular Solution: dy dx = f(x), y(x 0) = y 0 Examples: 1) dy dx = (x 2)2 ;y(2) = 1; 2) dy ;y(0) = 0; 3) dx

More information

Solution: APPM 2360 Exam1 Spring 2012

Solution: APPM 2360 Exam1 Spring 2012 APPM 60 Exam1 Spring 01 Problem 1: (0 points) Consider the following initial value problem where y is the radius of a raindrop and y = y / ; y 0 (1) y(0) = 1, () models the growth of the radius (as water

More information

Higher Order Linear Differential Equations with Constant Coefficients

Higher Order Linear Differential Equations with Constant Coefficients Higher Order Linear Differential Equations with Constant Coefficients Part I. Homogeneous Equations: Characteristic Roots Objectives: Solve n-th order homogeneous linear equations where a n,, a 1, a 0

More information

Math 267 - Practice exam 2 - solutions

Math 267 - Practice exam 2 - solutions C Roettger, Fall 13 Math 267 - Practice exam 2 - solutions Problem 1 A solution of 10% perchlorate in water flows at a rate of 8 L/min into a tank holding 200L pure water. The solution is kept well stirred

More information

Second-Order Linear Differential Equations

Second-Order Linear Differential Equations Second-Order Linear Differential Equations A second-order linear differential equation has the form 1 Px d 2 y dx 2 dy Qx dx Rxy Gx where P, Q, R, and G are continuous functions. We saw in Section 7.1

More information

Non-Homogeneous Equations

Non-Homogeneous Equations Non-Homogeneous Equations We now turn to finding solutions of a non-homogeneous second order linear equation. 1. Non-Homogeneous Equations 2. The Method of Undetermined Coefficients 3. The Method of Variation

More information

Absolute Maxima and Minima

Absolute Maxima and Minima Absolute Maxima and Minima Definition. A function f is said to have an absolute maximum on an interval I at the point x 0 if it is the largest value of f on that interval; that is if f( x ) f() x for all

More information

Sample Solutions of Assignment 5 for MAT3270B: Section 3.4. In each of problems find the general solution of the given differential equation

Sample Solutions of Assignment 5 for MAT3270B: Section 3.4. In each of problems find the general solution of the given differential equation Sample Solutions of Assignment 5 for MAT370B: 3.4-3.9 Section 3.4 In each of problems find the general solution of the given differential equation 7. y y + y = 0 1. 4y + 9y = 0 14. 9y + 9y 4y = 0 15. y

More information

Math 308 Week 1 Solutions

Math 308 Week 1 Solutions Math 308 Week 1 Solutions Here are solutions to the even-numbered suggested problems. The answers to the oddnumbered problems are in the back of your textbook, and the solutions are in the Solution Manual,

More information

Mechanical Vibrations

Mechanical Vibrations Mechanical Vibrations A mass m is suspended at the end of a spring, its weight stretches the spring by a length L to reach a static state (the equilibrium position of the system). Let u(t) denote the displacement,

More information

College of the Holy Cross, Spring 2009 Math 373, Partial Differential Equations Midterm 1 Practice Questions

College of the Holy Cross, Spring 2009 Math 373, Partial Differential Equations Midterm 1 Practice Questions College of the Holy Cross, Spring 29 Math 373, Partial Differential Equations Midterm 1 Practice Questions 1. (a) Find a solution of u x + u y + u = xy. Hint: Try a polynomial of degree 2. Solution. Use

More information

A 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 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 information

AP Calculus BC 2004 Free-Response Questions

AP Calculus BC 2004 Free-Response Questions AP Calculus BC 004 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 information

Math 410(510) Notes (4) Qualitative behavior of Linear systems a complete list. Junping Shi

Math 410(510) Notes (4) Qualitative behavior of Linear systems a complete list. Junping Shi A. Two different positive eigenvalues Math 4(5 Notes (4 Qualitative behavior of Linear systems a complete list Junping Shi Eigenvalues: λ = and λ = 4 Eigenvectors: V = (,, V = (, ( ( General solution:

More information

Homework #1 Solutions

Homework #1 Solutions MAT 303 Spring 203 Homework # Solutions Problems Section.:, 4, 6, 34, 40 Section.2:, 4, 8, 30, 42 Section.4:, 2, 3, 4, 8, 22, 24, 46... Verify that y = x 3 + 7 is a solution to y = 3x 2. Solution: From

More information

System of First Order Differential Equations

System 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 information

Math 20D Midterm Exam Practice Problems Solutions

Math 20D Midterm Exam Practice Problems Solutions Math 20D Midterm Exam Practice Problems Solutions 1. A tank contains G gallons of fresh water. A solution with a concentration of m lb of salt per gallon is pumped into the tank at a rate of r gallons

More information

MATH 215/255 Fall 2014 Assignment 2

MATH 215/255 Fall 2014 Assignment 2 MATH 215/255 Fall 214 Assignment 2 1.4, Exact equations ([Braun s Section 1.9]), 1.6 Solutions to selected exercises can be found in [Lebl], starting from page 33. 1.4.8: Solve 1 x 2 + 1 y + xy = 3 with

More information

Chapter (AB/BC, non-calculator) (a) Write an equation of the line tangent to the graph of f at x 2.

Chapter (AB/BC, non-calculator) (a) Write an equation of the line tangent to the graph of f at x 2. Chapter 1. (AB/BC, non-calculator) Let f( x) x 3 4. (a) Write an equation of the line tangent to the graph of f at x. (b) Find the values of x for which the graph of f has a horizontal tangent. (c) Find

More information

Math 2280 - Assignment 6

Math 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 information

(4.8) Solving Systems of Linear DEs by Elimination

(4.8) Solving Systems of Linear DEs by Elimination INTRODUCTION: (4.8) Solving Systems of Linear DEs by Elimination Simultaneous dinary differential equations involve two me equations that contain derivatives of two me dependent variables the unknown functions

More information

The integrating factor method (Sect. 2.1).

The 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 information

Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients. y + p(t) y + q(t) y = g(t), g(t) 0.

Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients. y + p(t) y + q(t) y = g(t), g(t) 0. Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients We will now turn our attention to nonhomogeneous second order linear equations, equations with the standard

More information

Math 209 Solutions to Assignment 7. x + 2y. 1 x + 2y i + 2. f x = cos(y/z)), f y = x z sin(y/z), f z = xy z 2 sin(y/z).

Math 209 Solutions to Assignment 7. x + 2y. 1 x + 2y i + 2. f x = cos(y/z)), f y = x z sin(y/z), f z = xy z 2 sin(y/z). Math 29 Solutions to Assignment 7. Find the gradient vector field of the following functions: a fx, y lnx + 2y; b fx, y, z x cosy/z. Solution. a f x x + 2y, f 2 y x + 2y. Thus, the gradient vector field

More information

A First Course in Elementary Differential Equations: Problems and Solutions. Marcel B. Finan Arkansas Tech University c All Rights Reserved

A First Course in Elementary Differential Equations: Problems and Solutions. Marcel B. Finan Arkansas Tech University c All Rights Reserved A First Course in Elementary Differential Equations: Problems and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved 1 Contents 1 Basic Terminology 4 2 Qualitative Analysis: Direction

More information

Area Between Curves. The idea: the area between curves y = f(x) and y = g(x) (if the graph of f(x) is above that of g(x)) for a x b is given by

Area Between Curves. The idea: the area between curves y = f(x) and y = g(x) (if the graph of f(x) is above that of g(x)) for a x b is given by MATH 42, Fall 29 Examples from Section, Tue, 27 Oct 29 1 The First Hour Area Between Curves. The idea: the area between curves y = f(x) and y = g(x) (if the graph of f(x) is above that of g(x)) for a x

More information

Homework #7 Solutions

Homework #7 Solutions MAT 0 Spring 201 Problems Homework #7 Solutions Section.: 4, 18, 22, 24, 4, 40 Section.4: 4, abc, 16, 18, 22. Omit the graphing part on problems 16 and 18...4. Find the general solution to the differential

More information

1. [20 pts] Find an integrating factor and solve the equation y 3y = e 2t. Then solve the initial value problem y 3y = e 2t, y(0) = 3.

1. [20 pts] Find an integrating factor and solve the equation y 3y = e 2t. Then solve the initial value problem y 3y = e 2t, y(0) = 3. 22M:034 Engineer Math IV: Differential Equations Midterm Exam 1 October 2, 2013 Name Section number 1. [20 pts] Find an integrating factor and solve the equation 3 = e 2t. Then solve the initial value

More information

Applications of Second-Order Differential Equations

Applications 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 information

Apr 23, 2015. Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23, 2015 sketching 1 / and 19pa

Apr 23, 2015. Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23, 2015 sketching 1 / and 19pa Calculus with Algebra and Trigonometry II Lecture 23 Final Review: Curve sketching and parametric equations Apr 23, 2015 Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23,

More information

General Theory of Differential Equations Sections 2.8, 3.1-3.2, 4.1

General Theory of Differential Equations Sections 2.8, 3.1-3.2, 4.1 A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics General Theory of Differential Equations Sections 2.8, 3.1-3.2, 4.1 Dr. John Ehrke Department of Mathematics Fall 2012 Questions

More information

Numerical Methods for Differential Equations

Numerical 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 information

Ordinary Differential Equations

Ordinary Differential Equations Ordinary Differential Equations Dan B. Marghitu and S.C. Sinha 1 Introduction An ordinary differential equation is a relation involving one or several derivatives of a function y(x) with respect to x.

More information

So far, we have looked at homogeneous equations

So far, we have looked at homogeneous equations Chapter 3.6: equations Non-homogeneous So far, we have looked at homogeneous equations L[y] = y + p(t)y + q(t)y = 0. Homogeneous means that the right side is zero. Linear homogeneous equations satisfy

More information

Higher Order Equations

Higher Order Equations Higher Order Equations We briefly consider how what we have done with order two equations generalizes to higher order linear equations. Fortunately, the generalization is very straightforward: 1. Theory.

More information

3.2 Sources, Sinks, Saddles, and Spirals

3.2 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 information

19.6. Finding a Particular Integral. Introduction. Prerequisites. Learning Outcomes. Learning Style

19.6. Finding a Particular Integral. Introduction. Prerequisites. Learning Outcomes. Learning Style Finding a Particular Integral 19.6 Introduction We stated in Block 19.5 that the general solution of an inhomogeneous equation is the sum of the complementary function and a particular integral. We have

More information

Math 41: Calculus Final Exam December 7, 2009

Math 41: Calculus Final Exam December 7, 2009 Math 41: Calculus Final Exam December 7, 2009 Name: SUID#: Select your section: Atoshi Chowdhury Yuncheng Lin Ian Petrow Ha Pham Yu-jong Tzeng 02 (11-11:50am) 08 (10-10:50am) 04 (1:15-2:05pm) 03 (11-11:50am)

More information

Math 432 HW 2.5 Solutions

Math 432 HW 2.5 Solutions Math 432 HW 2.5 Solutions Assigned: 1-10, 12, 13, and 14. Selected for Grading: 1 (for five points), 6 (also for five), 9, 12 Solutions: 1. (2y 3 + 2y 2 ) dx + (3y 2 x + 2xy) dy = 0. M/ y = 6y 2 + 4y N/

More information

19.7. Applications of Differential Equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

19.7. 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 information

A Brief Review of Elementary Ordinary Differential Equations

A Brief Review of Elementary Ordinary Differential Equations 1 A Brief Review of Elementary Ordinary Differential Equations At various points in the material we will be covering, we will need to recall and use material normally covered in an elementary course on

More information

AP Calculus BC 2010 Free-Response Questions

AP Calculus BC 2010 Free-Response Questions AP Calculus BC 2010 Free-Response Questions The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded

More information

Lecture Notes Math 250: Ordinary Differential Equations

Lecture Notes Math 250: Ordinary Differential Equations Lecture Notes Math 250: Ordinary Differential Equations Wen Shen 20 NB! These notes are used by myself. They are provided to students as a supplement to the textbook. They can not substitute the textbook.

More information

Problems for Quiz 14

Problems for Quiz 14 Problems for Quiz 14 Math 3. Spring, 7. 1. Consider the initial value problem (IVP defined by the partial differential equation (PDE u t = u xx u x + u, < x < 1, t > (1 with boundary conditions and initial

More information

Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.

Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality. 8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent

More information

People s Physics book 3e Ch 25-1

People s Physics book 3e Ch 25-1 The Big Idea: In most realistic situations forces and accelerations are not fixed quantities but vary with time or displacement. In these situations algebraic formulas cannot do better than approximate

More information

SECOND ORDER (inhomogeneous)

SECOND ORDER (inhomogeneous) Differential Equations SECOND ORDER (inhomogeneous) Graham S McDonald A Tutorial Module for learning to solve 2nd order (inhomogeneous) differential equations Table of contents Begin Tutorial c 2004 g.s.mcdonald@salford.ac.uk

More information

Math 22B, Homework #8 1. y 5y + 6y = 2e t

Math 22B, Homework #8 1. y 5y + 6y = 2e t Math 22B, Homework #8 3.7 Problem # We find a particular olution of the ODE y 5y + 6y 2e t uing the method of variation of parameter and then verify the olution uing the method of undetermined coefficient.

More information

Lecture Notes for Math250: Ordinary Differential Equations

Lecture Notes for Math250: Ordinary Differential Equations Lecture Notes for Math250: Ordinary Differential Equations Wen Shen 2011 NB! These notes are used by myself. They are provided to students as a supplement to the textbook. They can not substitute the textbook.

More information

Linear Equations and Inequalities

Linear Equations and Inequalities Linear Equations and Inequalities Section 1.1 Prof. Wodarz Math 109 - Fall 2008 Contents 1 Linear Equations 2 1.1 Standard Form of a Linear Equation................ 2 1.2 Solving Linear Equations......................

More information

Numerical Solution of Differential Equations

Numerical 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 information

Linear and quadratic Taylor polynomials for functions of several variables.

Linear and quadratic Taylor polynomials for functions of several variables. ams/econ 11b supplementary notes ucsc Linear quadratic Taylor polynomials for functions of several variables. c 010, Yonatan Katznelson Finding the extreme (minimum or maximum) values of a function, is

More information

The Predator-Prey Equations. x = a x α xy y = c y + γ xy

The Predator-Prey Equations. x = a x α xy y = c y + γ xy The Predator-Prey Equations An application of the nonlinear system of differential equations in mathematical biology / ecology: to model the predator-prey relationship of a simple eco-system. Suppose in

More information

Example 1: Competing Species

Example 1: Competing Species Local Linear Analysis of Nonlinear Autonomous DEs Local linear analysis is the process by which we analyze a nonlinear system of differential equations about its equilibrium solutions (also known as critical

More information

LS.6 Solution Matrices

LS.6 Solution Matrices LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions

More information

Name: ID: Discussion Section:

Name: ID: Discussion Section: Math 28 Midterm 3 Spring 2009 Name: ID: Discussion Section: This exam consists of 6 questions: 4 multiple choice questions worth 5 points each 2 hand-graded questions worth a total of 30 points. INSTRUCTIONS:

More information

Figure 2.1: Center of mass of four points.

Figure 2.1: Center of mass of four points. Chapter 2 Bézier curves are named after their inventor, Dr. Pierre Bézier. Bézier was an engineer with the Renault car company and set out in the early 196 s to develop a curve formulation which would

More information

The Phase Plane. Phase portraits; type and stability classifications of equilibrium solutions of systems of differential equations

The Phase Plane. Phase portraits; type and stability classifications of equilibrium solutions of systems of differential equations The Phase Plane Phase portraits; type and stability classifications of equilibrium solutions of systems of differential equations Phase Portraits of Linear Systems Consider a systems of linear differential

More information

Math 241, Exam 1 Information.

Math 241, Exam 1 Information. Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

More information

1 2 3 1 1 2 x = + x 2 + x 4 1 0 1

1 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 information

SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS

SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS A second-order linear differential equation has the form 1 Px d y dx dy Qx dx Rxy Gx where P, Q, R, and G are continuous functions. Equations of this type arise

More information

Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices

Linear 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 information

Mark Howell Gonzaga High School, Washington, D.C.

Mark 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 information

Centripetal force, rotary motion, angular velocity, apparent force.

Centripetal force, rotary motion, angular velocity, apparent force. Related Topics Centripetal force, rotary motion, angular velocity, apparent force. Principle and Task A body with variable mass moves on a circular path with ad-justable radius and variable angular velocity.

More information

Math 128A Spring 2003 Week 10 Solutions

Math 128A Spring 2003 Week 10 Solutions Math 128A Spring 2003 Week 10 Solutions Burden & Faires 5.4: 1a, 3a, 11a, 15, 17 Burden & Faires 5.5: 1a the equation should be y = te 3t 2y), 5 Burden & Faires 5.4. Runge-Kutta Methods 1. Use the Modified

More information

The dynamic equation for the angular motion of the wheel is R w F t R w F w ]/ J w

The 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 information

Elementary Statistics. Scatter Plot, Regression Line, Linear Correlation Coefficient, and Coefficient of Determination

Elementary Statistics. Scatter Plot, Regression Line, Linear Correlation Coefficient, and Coefficient of Determination Scatter Plot, Regression Line, Linear Correlation Coefficient, and Coefficient of Determination What is a Scatter Plot? A Scatter Plot is a plot of ordered pairs (x, y) where the horizontal axis is used

More information

Worksheet for Week 1: Circles and lines

Worksheet for Week 1: Circles and lines Worksheet Math 124 Week 1 Worksheet for Week 1: Circles and lines This worksheet is a review of circles and lines, and will give you some practice with algebra and with graphing. Also, this worksheet introduces

More information

Chapter 1 Vectors, lines, and planes

Chapter 1 Vectors, lines, and planes Simplify the following vector expressions: 1. a (a + b). (a + b) (a b) 3. (a b) (a + b) Chapter 1 Vectors, lines, planes 1. Recall that cross product distributes over addition, so a (a + b) = a a + a b.

More information

Solving First Order Linear Equations

Solving First Order Linear Equations Solving First Order Linear Equations Today we will discuss how to solve a first order linear equation. Our technique will require close familiarity with the product and chain rules for derivatives, so

More information

Section 1. Movement. So if we have a function x = f(t) that represents distance as a function of time, then dx is

Section 1. Movement. So if we have a function x = f(t) that represents distance as a function of time, then dx is Worksheet 4.4 Applications of Integration Section 1 Movement Recall that the derivative of a function tells us about its slope. What does the slope represent? It is the change in one variable with respect

More information

Taylor s approximation in several variables.

Taylor s approximation in several variables. ucsc supplementary notes ams/econ 11b Taylor s approximation in several variables. c 008, Yonatan Katznelson Finding the extreme (minimum or maximum values of a function, is one of the most important applications

More information

Homework 2 Solutions

Homework 2 Solutions Homework Solutions Igor Yanovsky Math 5B TA Section 5.3, Problem b: Use Taylor s method of order two to approximate the solution for the following initial-value problem: y = + t y, t 3, y =, with h = 0.5.

More information

Numerical Methods for Differential Equations

Numerical Methods for Differential Equations Numerical Methods for Differential Equations Course objectives and preliminaries Gustaf Söderlind and Carmen Arévalo Numerical Analysis, Lund University Textbooks: A First Course in the Numerical Analysis

More information

Readings this week. 1 Parametric Equations Supplement. 2 Section 10.1. 3 Sections 2.1-2.2. Professor Christopher Hoffman Math 124

Readings this week. 1 Parametric Equations Supplement. 2 Section 10.1. 3 Sections 2.1-2.2. Professor Christopher Hoffman Math 124 Readings this week 1 Parametric Equations Supplement 2 Section 10.1 3 Sections 2.1-2.2 Precalculus Review Quiz session Thursday equations of lines and circles worksheet available at http://www.math.washington.edu/

More information

9. Particular Solutions of Non-homogeneous second order equations Undetermined Coefficients

9. Particular Solutions of Non-homogeneous second order equations Undetermined Coefficients September 29, 201 9-1 9. Particular Solutions of Non-homogeneous second order equations Undetermined Coefficients We have seen that in order to find the general solution to the second order differential

More information

CITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION

CITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August

More information

P(a X b) = f X (x)dx. A p.d.f. must integrate to one: f X (x)dx = 1. Z b

P(a X b) = f X (x)dx. A p.d.f. must integrate to one: f X (x)dx = 1. Z b Continuous Random Variables The probability that a continuous random variable, X, has a value between a and b is computed by integrating its probability density function (p.d.f.) over the interval [a,b]:

More information

Simple Harmonic Motion

Simple Harmonic Motion Simple Harmonic Motion 9M Object: Apparatus: To determine the force constant of a spring and then study the harmonic motion of that spring when it is loaded with a mass m. Force sensor, motion sensor,

More information

AP Calculus AB 2005 Free-Response Questions

AP Calculus AB 2005 Free-Response Questions AP Calculus AB 25 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 information

HOMEWORK 4 SOLUTIONS. All questions are from Vector Calculus, by Marsden and Tromba

HOMEWORK 4 SOLUTIONS. All questions are from Vector Calculus, by Marsden and Tromba HOMEWORK SOLUTIONS All questions are from Vector Calculus, by Marsden and Tromba Question :..6 Let w = f(x, y) be a function of two variables, and let x = u + v, y = u v. Show that Solution. By the chain

More information

Block 7: Applications of Differentiation Contents

Block 7: Applications of Differentiation Contents Block 7: Applications of Differentiation Contents 7.1 Optimisation... 2 7.2 Velocity and Acceleration... 4 7.3 Differential Equations... 6 7.4 First Order Differential Equations... 7 7.5 Second Order Differential

More information

r (t) = 2r(t) + sin t θ (t) = r(t) θ(t) + 1 = 1 1 θ(t) 1 9.4.4 Write the given system in matrix form x = Ax + f ( ) sin(t) x y 1 0 5 z = dy cos(t)

r (t) = 2r(t) + sin t θ (t) = r(t) θ(t) + 1 = 1 1 θ(t) 1 9.4.4 Write the given system in matrix form x = Ax + f ( ) sin(t) x y 1 0 5 z = dy cos(t) Solutions HW 9.4.2 Write the given system in matrix form x = Ax + f r (t) = 2r(t) + sin t θ (t) = r(t) θ(t) + We write this as ( ) r (t) θ (t) = ( ) ( ) 2 r(t) θ(t) + ( ) sin(t) 9.4.4 Write the given system

More information

Notes from FE Review for mathematics, University of Kentucky

Notes from FE Review for mathematics, University of Kentucky Notes from FE Review for mathematics, University of Kentucky Dr. Alan Demlow March 19, 2012 Introduction These notes are based on reviews for the mathematics portion of the Fundamentals of Engineering

More information

PSTricks. pst-ode. A PSTricks package for solving initial value problems for sets of Ordinary Differential Equations (ODE), v0.7.

PSTricks. pst-ode. A PSTricks package for solving initial value problems for sets of Ordinary Differential Equations (ODE), v0.7. PSTricks pst-ode A PSTricks package for solving initial value problems for sets of Ordinary Differential Equations (ODE), v0.7 27th March 2014 Package author(s): Alexander Grahn Contents 2 Contents 1 Introduction

More information

The graphs of f and g intersect at (0, 0) and one other point. Find that point: f(y) = g(y) y 2 4y 2y 2 6y = = 2y y 2. 2y(y 3) = 0

The graphs of f and g intersect at (0, 0) and one other point. Find that point: f(y) = g(y) y 2 4y 2y 2 6y = = 2y y 2. 2y(y 3) = 0 . Compute the area between the curves x y 4y and x y y. Let f(y) y 4y y(y 4). f(y) when y or y 4. Let g(y) y y y( y). g(y) when y or y. x 3 y? The graphs of f and g intersect at (, ) and one other point.

More information

1. (1 pt) Write the given second order equation as its equivalent system of first order equations. u =

1. (1 pt) Write the given second order equation as its equivalent system of first order equations. u = 1 (1 pt) Write the given second order equation as its equivalent system of first order equations u + u + 5u = 0 Use v to represent the velocity function, ie v = u (t) Use v and u for the two functions,

More information

Math 201 Lecture 23: Power Series Method for Equations with Polynomial

Math 201 Lecture 23: Power Series Method for Equations with Polynomial Math 201 Lecture 23: Power Series Method for Equations with Polynomial Coefficients Mar. 07, 2012 Many examples here are taken from the textbook. The first number in () refers to the problem number in

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