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.


 April Shepherd
 3 years ago
 Views:
Transcription
1 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 problem 3 = e 2t, (0) = 3. Solution: We can use the integrating factor µ(t) = e 3t to convert the original equation into e 3t 3e 3t = e 3t e 2t or (e 3t ) = e t. Then, b integrating both sides of this equation and solving for we obtain the general solution of 3 = e 2t, namel = Ce 3t e 2t where C is an arbitrar constant. To satisf the condition (0) = 3, we must choose C = 4. Thus the unique solution of the initial value problem is = 4e 3t e 2t.
2 2. [20 pts] Solve the initial value problem dx = 2x 2 + 1, (2) = 0 and detere the interval in which the solution exists. Solution: The differential equation can be written as (2 + 1) = 2xdx. Integrating the left side with respect to and the right side with respect to x gives 2 + = x 2 + C, where C is an arbitrar constant. To satisf (2) = 0 we must have (2) 2 + (2) = C, C = 4. Hence the solution of the initial value problem is given implicitl b 2 + = x 2 4. To obtain the solution explicitl, we solve the above equation for in terms of x and we obtain = ± 1 2 4x This gives two solutions of the original differential equation, onl one of which, however, satisfies the given initial condition. This is the solution corresponding to the plus sign; thus we obtain = 1 2 ( 4x ) as the solution of the initial value problem. Finall, to detere the interval in which this solution is valid, we must find the interval containing 2 in which the quantit under the radical is positive. We find that the desired interval is x > 15 2.
3 3. [20 pts] At time t = 0 a tank contains Q 0 of salt dissolved in 100 gal of water. Assume that water containing 1 of salt per gallon 4 is entering the tank at a rate of 3 gal/, and that the wellstirred solution is leaving the tank at the same rate. Find an expression for the amount of salt Q(t) in the tank at an time t. Solution: Rate of change Rate at which Rate at which of amount of salt salt is flowing in salt is flowing out in tank in in in dq dt = Q(t) gal gal gal gal The problem ields a separable firstorder linear differential equation with initial condition Q (t) = Q(t), Q(0) = Q 0. The general solution of the differential equation is Q(t) = Ce 3t , where C is an arbitrar constant. To satisf the initial condition, we must choose C = Q Thus the unique solution of the initial value problem, Q(t) = (Q 0 25)e 3t , gives the expression for the amount of salt Q(t) in the tank at an time t.
4 4. [20 pts] Solve the differential equation (3x )dx + (x 3 + 2x + e ) = 0. Solution: Let M(x, ) = 3x and N(x, ) = x 3 + 2x + e. Then M = N 3x2 + 2 and x = 3x2 + 2, and since these are equal, the equation is exact. Thus there is a potential function ψ = ψ(x, ) such that ψ x (x, ) = 3x2 + 2 ψ (x, ) = x3 + 2x + e. Our goal is to solve this sstem of equations for the function ψ(x, ). The first of these equations can be integrated with respect to x (holding constant) to find ψ(x, ) = x 3 + x 2 + h(), where the function h is an arbitrar differentiable function of, plaing the role of an arbitrar constant. B differentiating the above equation with respect to and comparing the result with the second equation in the sstem, we obtain x 3 + 2x + h () = x 3 + 2x + e, h () = e, and as a solution of the above separable equation we can take h() = e (we do not require the most general one). Finall, solutions of the original differential equation, (3x )dx + (x 3 + 2x + e ) = 0, are given implicitl b ψ(x, ) = C or x 3 + x 2 + e = C, where C is an arbitrar constant.
5 5. [20 pts] Solve the equation = (2 ). dx Also find the equilibrium solutions, classif each one as asmptoticall stable or unstable, and sketch the equilibrium solutions and several other solutions in the xplane. Solution: To solve the equation, we integrate both sides of (2 ) and we obtain: (2 ) = dx ( ) = x + C = x + C ln 1 2 ln 2 = x + C 1 ln ln 2 = 2x + 2C 1 ln 2 = 2x + C 2 (C 2 = 2C 1 ) e ln 2 = e 2x+C 2 2 = e2x e C 2 (e C 2 > 0) = Ce 2x (C R) 2 = Ce 2x (2 ) = 2Ce 2x Ce 2x + Ce 2x = 2Ce 2x (1 + Ce 2x ) = 2Ce 2x Thus, the solutions are given b = 2Ce 2x 1 + Ce 2x. (x) = 2e2x c + e. 2x (c = 1 ) C There are two equilibrium solutions to this equation = 0 unstable, = 2 stable. We sketch the equilibrium solutions and several other solutions on the next page.
Autonomous 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 informationHomework #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 informationParticular Solutions. y = Ae 4x and y = 3 at x = 0 3 = Ae 4 0 3 = A y = 3e 4x
Particular Solutions If the differential equation is actually modeling something (like the cost of milk as a function of time) it is likely that you will know a specific value (like the fact that milk
More information15.1. Exact Differential Equations. Exact FirstOrder Equations. Exact Differential Equations Integrating Factors
SECTION 5. Eact FirstOrder Equations 09 SECTION 5. Eact FirstOrder Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential
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 closedbook test, and is to be taken without the use
More informationSection 7.2 Linear Programming: The Graphical Method
Section 7.2 Linear Programming: The Graphical Method Man problems in business, science, and economics involve finding the optimal value of a function (for instance, the maimum value of the profit function
More informationC.4 Applications of Differential Equations
APPENDIX C Differential Equations A39 C.4 Applications of Differential Equations Use differential equations to model and solve reallife problems. EXAMPLE 1 Modeling Advertising Awareness The new cereal
More informationApplications of 1stOrder Equations
3 Applications of 1stOrder Equations 3.2 Compartmental Analysis For a first example we shall analyze a onecompartment system, and then later consider a twocompartment system. Example 3.1. A brine solution
More informationMath 432 HW 2.5 Solutions
Math 432 HW 2.5 Solutions Assigned: 110, 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 informationMath 115 HW #8 Solutions
Math 115 HW #8 Solutions 1 The function with the given graph is a solution of one of the following differential equations Decide which is the correct equation and justify your answer a) y = 1 + xy b) y
More informationMATH 2300 review problems for Exam 3 ANSWERS
MATH 300 review problems for Exam 3 ANSWERS. Check whether the following series converge or diverge. In each case, justify your answer by either computing the sum or by by showing which convergence test
More informationPractice Material for Final Differential Equations
Practice Material for Final Differential Equations Note that this list of questions is not ehaustive. You should also be prepared to answer questions such as those on test, test, the make up test for test,
More information15.2. FirstOrder Linear Differential Equations. FirstOrder Linear Differential Equations Bernoulli Equations Applications
00 CHAPTER 5 Differential Equations SECTION 5. FirstOrer Linear Differential Equations FirstOrer Linear Differential Equations Bernoulli Equations Applications FirstOrer Linear Differential Equations
More informationy cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx
Trigonometric Integrals In this section we use trigonometric identities to integrate certain combinations of trigonometric functions. We start with powers of sine and cosine. EXAMPLE Evaluate cos 3 x dx.
More informationECG590I Asset Pricing. Lecture 2: Present Value 1
ECG59I Asset Pricing. Lecture 2: Present Value 1 2 Present Value If you have to decide between receiving 1$ now or 1$ one year from now, then you would rather have your money now. If you have to decide
More informationMath 115 HW #9 Solutions
1. Solve the differential equation Math 115 HW #9 Solutions sin x dy dx + (os x)y = x sin(x2 ). Answer: Dividing everything by sin x yields the linear first order equation Here P (x) = os x sin x and Q(x)
More information1. (from Stewart, page 586) Solve the initial value problem.
. (from Stewart, page 586) Solve the initial value problem.. (from Stewart, page 586) (a) Solve y = y. du dt = t + sec t u (b) Solve y = y, y(0) = 0., u(0) = 5. (c) Solve y = y, y(0) = if possible. 3.
More informationHW 3 Due Sep 12, Wed
HW 3 Due Sep, Wed 1. Consider a tan used in certain hydrodynamic experiments. After one experiment the tan contains 200 L of a dye solution with a concentration of 1 g/l. To prepare for the next experiment,
More informationGraphing Linear Inequalities in Two Variables
5.4 Graphing Linear Inequalities in Two Variables 5.4 OBJECTIVES 1. Graph linear inequalities in two variables 2. Graph a region defined b linear inequalities What does the solution set look like when
More informationBlock 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 informationMath 370, Spring 2008 Prof. A.J. Hildebrand. Practice Test 1 Solutions
Math 70, Spring 008 Prof. A.J. Hildebrand Practice Test Solutions About this test. This is a practice test made up of a random collection of 5 problems from past Course /P actuarial exams. Most of the
More informationMath 370/408, Spring 2008 Prof. A.J. Hildebrand. Actuarial Exam Practice Problem Set 3 Solutions
Math 37/48, Spring 28 Prof. A.J. Hildebrand Actuarial Exam Practice Problem Set 3 Solutions About this problem set: These are problems from Course /P actuarial exams that I have collected over the years,
More information1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,
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 informationCh 8 Potential energy and Conservation of Energy. Question: 2, 3, 8, 9 Problems: 3, 9, 15, 21, 24, 25, 31, 32, 35, 41, 43, 47, 49, 53, 55, 63
Ch 8 Potential energ and Conservation of Energ Question: 2, 3, 8, 9 Problems: 3, 9, 15, 21, 24, 25, 31, 32, 35, 41, 43, 47, 49, 53, 55, 63 Potential energ Kinetic energ energ due to motion Potential energ
More information7. Continuously Varying Interest Rates
7. Continuously Varying Interest Rates 7.1 The Continuous Varying Interest Rate Formula. Suppose that interest is continuously compounded with a rate which is changing in time. Let the present time be
More information(amount of salt in) (amount of salt out),
MATH 222 (Lectures 1,2,4) Worksheet 8.5 Solutions Please inform your TA if you find any errors in the solutions. 1. A 100 gallon tank is full of pure water. Let pure water run into the tank at the rate
More informationProblem Set 3 Solutions
Chemistry 360 Dr Jean M Standard Problem Set 3 Solutions 1 (a) One mole of an ideal gas at 98 K is expanded reversibly and isothermally from 10 L to 10 L Determine the amount of work in Joules We start
More information3. Which of the following couldn t be the solution of a differential equation? (a) z(t) = 6
MathQuest: Differential Equations What is a Differential Equation? 1. Which of the following is not a differential equation? (a) y = 3y (b) 2x 2 y + y 2 = 6 (c) tx dx dt = 2 (d) d2 y + 4 dy dx 2 dx + 7y
More informationSolutions to Homework Section 3.7 February 18th, 2005
Math 54W Spring 5 Solutions to Homeork Section 37 Februar 8th, 5 List the ro vectors and the column vectors of the matrix The ro vectors are The column vectors are ( 5 5 The matrix ( (,,,, 4, (5,,,, (
More informationMath 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 informationy cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx y 1 u 2 du u 1 3u 3 C
Trigonometric Integrals In this section we use trigonometric identities to integrate certain combinations of trigonometric functions. We start with powers of sine and cosine. EXAMPLE Evaluate cos 3 x dx.
More informationUNIVERSITY OF CALIFORNIA, SAN DIEGO Electrical & Computer Engineering Department ECE 101  Fall 2009 Linear Systems Fundamentals
UNIVERSITY OF CALIFORNIA, SAN DIEGO Electrical & Computer Engineering Department ECE 101  Fall 2009 Linear Systems Fundamentals MIDTERM EXAM You are allowed one 2sided sheet of notes. No books, no other
More informationAlgebra II Notes Piecewise Functions Unit 1.5. Piecewise linear functions. Math Background
Piecewise linear functions Math Background Previousl, ou Related a table of values to its graph. Graphed linear functions given a table or an equation. In this unit ou will Determine when a situation requiring
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 informationSolutions to Linear First Order ODE s
First Order Linear Equations In the previous session we learned that a first order linear inhomogeneous ODE for the unknown function x = x(t), has the standard form x + p(t)x = q(t) () (To be precise we
More informationTo Be or Not To Be a Linear Equation: That Is the Question
To Be or Not To Be a Linear Equation: That Is the Question Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form A + B C where A and B are not
More informationPartial differentiation
Chapter 1 Partial differentiation Example 1.1 What is the maximal domain of the real function g defined b g(x) = x 2 + 3x + 2? : The ke point is that the square root onl gives a real result if the argument
More informationNonhomogeneous Linear Equations
Nonhomogeneous Linear Equations In this section we learn how to solve secondorder nonhomogeneous linear differential equations with constant coefficients, that is, equations of the form ay by cy G x where
More informationFIRSTORDER DIFFERENTIAL EQUATIONS
Chapter 16 FIRSTORDER DIFFERENTIAL EQUATIONS OVERVIEW In Section 4.7 we introduced differential euations of the form >d = ƒ(), where ƒ is given and is an unknown function of. When ƒ is continuous over
More information2.6. The Circle. Introduction. Prerequisites. Learning Outcomes
The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures and has been around a long time! In this brief Section we discuss the basic coordinate geometr of a circle  in particular
More informationMath 425 (Fall 08) Solutions Midterm 2 November 6, 2008
Math 425 (Fall 8) Solutions Midterm 2 November 6, 28 (5 pts) Compute E[X] and Var[X] for i) X a random variable that takes the values, 2, 3 with probabilities.2,.5,.3; ii) X a random variable with the
More informationMATHEMATICS Extended Part Module 1 (Calculus and Statistics) (Sample Paper)
HONG KONG EXAMINATIONS AND ASSESSMENT AUTHORITY HONG KONG DIPLOMA OF SECONDARY EDUCATION EXAMINATION MATHEMATICS Extended Part Module 1 (Calculus and Statistics) (Sample Paper) Time allowed: hours 30 minutes
More informationFirst Order NonLinear Equations
First Order NonLinear Equations We will briefly consider nonlinear 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 informationAlgebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test
Dear Parents, Based on the results of the High School Placement Test (HSPT), your child should forecast to take Algebra 1 this fall. If you are okay with that placement then you have no further action
More informationChapter 7 Outline Math 236 Spring 2001
Chapter 7 Outline Math 236 Spring 2001 Note 1: Be sure to read the Disclaimer on Chapter Outlines! I cannot be responsible for misfortunes that may happen to you if you do not. Note 2: Section 7.9 will
More information1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style
Factorisation 1.5 Introduction In Block 4 we showed the way in which brackets were removed from algebraic expressions. Factorisation, which can be considered as the reverse of this process, is dealt with
More information1.(6pts) Find symmetric equations of the line L passing through the point (2, 5, 1) and perpendicular to the plane x + 3y z = 9.
.(6pts Find symmetric equations of the line L passing through the point (, 5, and perpendicular to the plane x + 3y z = 9. (a x = y + 5 3 = z (b x (c (x = ( 5(y 3 = z + (d x (e (x + 3(y 3 (z = 9 = y 3
More information1 Maximizing pro ts when marginal costs are increasing
BEE12 Basic Mathematical Economics Week 1, Lecture Tuesda 12.1. Pro t maimization 1 Maimizing pro ts when marginal costs are increasing We consider in this section a rm in a perfectl competitive market
More informationMath 370, Spring 2008 Prof. A.J. Hildebrand. Practice Test 2 Solutions
Math 370, Spring 008 Prof. A.J. Hildebrand Practice Test Solutions About this test. This is a practice test made up of a random collection of 5 problems from past Course /P actuarial exams. Most of the
More information2. Perform the division as if the numbers were whole numbers. You may need to add zeros to the back of the dividend to complete the division
Math Section 5. Dividing Decimals 5. Dividing Decimals Review from Section.: Quotients, Dividends, and Divisors. In the expression,, the number is called the dividend, is called the divisor, and is called
More informationMath 2280  Assignment 6
Math 2280  Assignment 6 Dylan Zwick Spring 2014 Section 3.81, 3, 5, 8, 13 Section 4.11, 2, 13, 15, 22 Section 4.21, 10, 19, 28 1 Section 3.8  Endpoint Problems and Eigenvalues 3.8.1 For the eigenvalue
More informationDimensional Analysis is a simple method for changing from one unit of measure to another. How many yards are in 49 ft?
HFCC Math Lab NAT 05 Dimensional Analysis Dimensional Analysis is a simple method for changing from one unit of measure to another. Can you answer these questions? How many feet are in 3.5 yards? Revised
More informationMATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4.
MATH 55 SOLUTIONS TO PRACTICE FINAL EXAM x 2 4.Compute x 2 x 2 5x + 6. When x 2, So x 2 4 x 2 5x + 6 = (x 2)(x + 2) (x 2)(x 3) = x + 2 x 3. x 2 4 x 2 x 2 5x + 6 = 2 + 2 2 3 = 4. x 2 9 2. Compute x + sin
More informationReview for Exam 2 (The following problems are all from old exams)
Review for Exam 2 (The following problems are all from old exams). Write down, but do not evaluate, an integral which represents the volume when y = e x, x, is rotated about the y axis. Soln: Divide the
More information5 Indefinite integral
5 Indefinite integral The most of the mathematical operations have inverse operations: the inverse operation of addition is subtraction, the inverse operation of multiplication is division, the inverse
More informationChapter 11  Curve Sketching. Lecture 17. MATH10070  Introduction to Calculus. maths.ucd.ie/modules/math10070. Kevin Hutchinson.
Lecture 17 MATH10070  Introduction to Calculus maths.ucd.ie/modules/math10070 Kevin Hutchinson 28th October 2010 Z Chain Rule (I): If y = f (u) and u = g(x) dy dx = dy du du dx Z Chain rule (II): d dx
More informationDon't Forget the Differential Equations: Finishing 2005 BC4
connect to college success Don't Forget the Differential Equations: Finishing 005 BC4 Steve Greenfield available on apcentral.collegeboard.com connect to college success www.collegeboard.com The College
More information1. Firstorder Ordinary Differential Equations
Advanced Engineering Mathematics 1. Firstorder ODEs 1 1. Firstorder Ordinary Differential Equations 1.1 Basic concept and ideas 1.2 Geometrical meaning of direction fields 1.3 Separable differential
More informationStudent Activity: To investigate the Average Value of a Function
Student Activity: To investigate the Average Value of a Function Use in connection with the interactive file, Average Value 3, on the Student s CD. 1. Click all the boxes in the interactive file. Move
More informationFINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA
FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA 1.1 Solve linear equations and equations that lead to linear equations. a) Solve the equation: 1 (x + 5) 4 = 1 (2x 1) 2 3 b) Solve the equation: 3x
More informationVirtual Power Limiter System which Guarantees Stability of Control Systems
Virtual Power Limiter Sstem which Guarantees Stabilit of Control Sstems Katsua KANAOKA Department of Robotics, Ritsumeikan Universit Shiga 5258577, Japan Email: kanaoka@se.ritsumei.ac.jp Abstract In this
More information1.1 Exercises Answers to selected oddnumbered problems begin on page ANS1.
. Eercises Answers to selected oddnumbered problems begin on page ANS. In Problems 8, state the order of the given ordinar differential equation. Determine whether the equation is linear or nonlinear
More informationWS 7.5: Partial Fractions & Logistic. Name Date Period
Name Date Period Worksheet 7.5 Partial Fractions & Logistic Growth Show all work. No calculator unless stated. Multiple Choice 1. The spread of a disease through a community can be modeled with the logistic
More information2 Applications to Business and Economics
2 Applications to Business and Economics APPLYING THE DEFINITE INTEGRAL 442 Chapter 6 Further Topics in Integration In Section 6.1, you saw that area can be expressed as the limit of a sum, then evaluated
More informationFlorida Algebra I EOC Online Practice Test
Florida Algebra I EOC Online Practice Test Directions: This practice test contains 65 multiplechoice questions. Choose the best answer for each question. Detailed answer eplanations appear at the end
More information2.6. The Circle. Introduction. Prerequisites. Learning Outcomes
The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures. In this brief Section we discuss the basic coordinate geometr of a circle  in particular the basic equation representing
More information2 Integrating Both Sides
2 Integrating Both Sides So far, the only general method we have for solving differential equations involves equations of the form y = f(x), where f(x) is any function of x. The solution to such an equation
More informationAP Calculus BC 2007 FreeResponse Questions
AP Calculus BC 7 FreeResponse Questions The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to college
More informationA Dynamic Analysis of Price Determination Under Joint Profit Maximization in Bilateral Monopoly
A Dynamic Analysis of Price Determination Under Joint Profit Maximization in Bilateral Monopoly by Stephen Devadoss Department of Agricultural Economics University of Idaho Moscow, Idaho 838442334 Phone:
More information1 Calculus of Several Variables
1 Calculus of Several Variables Reading: [Simon], Chapter 14, p. 30031. 1.1 Partial Derivatives Let f : R n R. Then for each x i at each point x 0 = (x 0 1,..., x 0 n) the ith partial derivative is defined
More informationAP Calculus AB 2003 Scoring Guidelines Form B
AP Calculus AB Scoring Guidelines Form B The materials included in these files are intended for use by AP teachers for course and exam preparation; permission for any other use must be sought from the
More informationFixed Point Theorems
Fixed Point Theorems Definition: Let X be a set and let T : X X be a function that maps X into itself. (Such a function is often called an operator, a transformation, or a transform on X, and the notation
More informationEASTERN ARIZONA COLLEGE Differential Equations
EASTERN ARIZONA COLLEGE Differential Equations Course Design 20152016 Course Information Division Mathematics Course Number MAT 260 (SUN# MAT 2262) Title Differential Equations Credits 3 Developed by
More informationPRACTICE FINAL. Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 10cm.
PRACTICE FINAL Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 1cm. Solution. Let x be the distance between the center of the circle
More informationProblem Solving 8: RC and LR Circuits
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Problem Solving 8: RC and LR Circuits Section Table and Group (e.g. L04 3C ) Names Hand in one copy per group at the end of the Friday Problem
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More information3 Rectangular Coordinate System and Graphs
060_CH03_13154.QXP 10/9/10 10:56 AM Page 13 3 Rectangular Coordinate Sstem and Graphs In This Chapter 3.1 The Rectangular Coordinate Sstem 3. Circles and Graphs 3.3 Equations of Lines 3.4 Variation Chapter
More informationHW 3 Due Sep 12, Wed
HW 3 Due Sep, Wed 1. Consider a tank used in certain hydrodynamic experiments. After one experiment the tank contains 200 L of a dye solution with a concentration of 1 g/l. To prepare for the next experiment,
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 notforprofit membership association whose mission is to connect students to college
More information2.4 Inequalities with Absolute Value and Quadratic Functions
08 Linear and Quadratic Functions. Inequalities with Absolute Value and Quadratic Functions In this section, not onl do we develop techniques for solving various classes of inequalities analticall, we
More informationP1. Plot the following points on the real. P2. Determine which of the following are solutions
Section 1.5 Rectangular Coordinates and Graphs of Equations 9 PART II: LINEAR EQUATIONS AND INEQUALITIES IN TWO VARIABLES 1.5 Rectangular Coordinates and Graphs of Equations OBJECTIVES 1 Plot Points in
More information19.4. Integrating Factor. Introduction. Prerequisites. Learning Outcomes. Learning Style
Integrating Factor 19.4 Introduction An exact differential equation is one which can be solved by simply integrating both sides. Whilst such equations are few and far between an important class of differential
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 informationSECOND 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 informationNew HigherProposed OrderCombined Approach. Block 1. Lines 1.1 App. Vectors 1.4 EF. Quadratics 1.1 RC. Polynomials 1.1 RC
New HigherProposed OrderCombined Approach Block 1 Lines 1.1 App Vectors 1.4 EF Quadratics 1.1 RC Polynomials 1.1 RC Differentiationbut not optimisation 1.3 RC Block 2 Functions and graphs 1.3 EF Logs
More informationdy dx and so we can rewrite the equation as If we now integrate both sides of this equation, we get xy x 2 C Integrating both sides, we would have
Linear Differential Equations A firstder linear differential equation is one that can be put into the fm 1 d P y Q where P and Q are continuous functions on a given interval. This type of equation occurs
More informationLecture VI. Review of even and odd functions Definition 1 A function f(x) is called an even function if. f( x) = f(x)
ecture VI Abstract Before learning to solve partial differential equations, it is necessary to know how to approximate arbitrary functions by infinite series, using special families of functions This process
More informationMATH 121 FINAL EXAM FALL 20102011. December 6, 2010
MATH 11 FINAL EXAM FALL 010011 December 6, 010 NAME: SECTION: Instructions: Show all work and mark your answers clearly to receive full credit. This is a closed notes, closed book exam. No electronic
More informationSample Midterm Solutions
Sample Midterm Solutions Instructions: Please answer both questions. You should show your working and calculations for each applicable problem. Correct answers without working will get you relatively few
More informationCollege 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 informationChapter 4 Expected Values
Chapter 4 Expected Values 4. The Expected Value of a Random Variables Definition. Let X be a random variable having a pdf f(x). Also, suppose the the following conditions are satisfied: x f(x) converges
More informationCoffeyville Community College #MATH 202 COURSE SYLLABUS FOR DIFFERENTIAL EQUATIONS. Ryan Willis Instructor
Coffeyville Community College #MATH 202 COURSE SYLLABUS FOR DIFFERENTIAL EQUATIONS Ryan Willis Instructor COURSE NUMBER: MATH 202 COURSE TITLE: Differential Equations CREDIT HOURS: 3 INSTRUCTOR: OFFICE
More informationAP Calculus AB 2007 FreeResponse Questions
AP Calculus AB 2007 FreeResponse Questions The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to
More informationA 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 informationSolving DEs by Separation of Variables.
Solving DEs by Separation of Variables. Introduction and procedure Separation of variables allows us to solve differential equations of the form The steps to solving such DEs are as follows: dx = gx).
More information3 Optimizing Functions of Two Variables. Chapter 7 Section 3 Optimizing Functions of Two Variables 533
Chapter 7 Section 3 Optimizing Functions of Two Variables 533 (b) Read about the principle of diminishing returns in an economics tet. Then write a paragraph discussing the economic factors that might
More informationSECONDORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS
L SECONDORDER LINEAR HOOGENEOUS DIFFERENTIAL EQUATIONS SECONDORDER LINEAR HOOGENEOUS DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS A secondorder linear differential equation is one of the form d
More informationMATH 461: Fourier Series and Boundary Value Problems
MATH 461: Fourier Series and Boundary Value Problems Chapter III: Fourier Series Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2015 fasshauer@iit.edu MATH 461 Chapter
More information