California State University Northridge MATH 280: Applied Differential Equations Midterm Exam 2
|
|
- Dinah Greer
- 6 years ago
- Views:
Transcription
1 California State University Northridge MATH 280: Applied Differential Equations Midterm Exam 2 November 3, 203. Duration: 75 Minutes. Instructor: Jing Li Student Name: Student number: Take your time to read the entire paper before you begin to write, and read each question carefully. Remember that certain questions are worth more points than others. Make a note of the questions that you feel confident you can do, and then do those first: you do not have to proceed through the paper in the order given. You have 75 minutes to complete this exam. This is a closed book exam, and no notes of any kind are allowed. The use of cell phones, pagers or any text storage or communication device is not permitted. Only the Faculty approved TI-30 calculator is allowed. The correct answer requires justification written legibly and logically: you must convince me that you know why your solution is correct. Answer these questions in the space provided. Use the backs of pages if necessary. Where it is possible to check your work, do so. Good Luck! Problem Total (BONUS) Points Your Marks
2 Question. [8 points] Determine whether the given set of fuctions is linearly independent on the interval (, ). (a) [4 points] f (x) = x, f 2 (x) = x 2, f 3 (x) = 4x 3x 2 Solution: Since f 3 (x) = 4 f (x) 3 f 2 (x) = 4 f (x) 3 f 2 (x) f 3 (x) = 0, the set of functions f (x), f 2 (x), f 3 (x) is linearly dependent. (b) [4 points] f (x) = + x, f 2 (x) = x, f 3 (x) = x 2 Solution: Let c f (x) + c 2 f 2 (x) + c 3 f 3 (x) = 0 = c ( + x) + c 2 x + c 3 x 2 = 0 = c + (c + c 2 )x + c 3 x 2 = 0, then c = 0,c + c 2 = 0,c 3 = 0,= c = c 2 = c 3 = 0. Hence the set of functions f (x), f 2 (x), f 3 (x) is linearly independent. 2
3 Question 2. [24 points] Consider the differential equation x 3 y + x 2 y 2xy + 2y = 0 (a) [5 points] Verify that the fuctions of y = x, y 2 = x 2, y 3 = x differential equation. Solution: are solutions of the given Check y is the sol of x 3 y + x 2 y 2xy + 2y = 0 y = x = y =, y = 0, y = 0 Then x 3 y + x2 y 2xy + 2y = x + 2x = 0. Check y 2 is the solution of x 3 y + x 2 y 2xy + 2y = 0 y 2 = x 2 = y 2 = 2x, y 2 = 2, y 2 = 0 Then x 3 y 2 + x2 y 2 2xy 2 + 2y 2 = 0 + x 2 2 2x 2x + 2x 2 = 0. Check y 3 is the solution of x 3 y + x 2 y 2xy + 2y = 0 y 3 = x = y 3 = x 2, y 3 = 2x 3, y 3 = 6x 4 Then x 3 y 3 + x2 y 3 2xy 3 + 2y 3 = 0. (b) [7 points] Verify that y, y 2, y 3 in part (a) form a fundamental set of solutions for the given differential equaiton on the interval of (0, ). Solution: W (y, y 2, y 3 )(x) = y y 2 y 3 y y2 y3 y y2 y3 = Then y, y 2, y 3 form a fundamental set of solutions. x x 2 x 2x x x 3 = 6 x = 0, on (0, ) (c) [2 points] Using y, y 2 and y 3 in part (a), form the general solution of the given differential equation. Solution: The general solution is y(x) = c y + c 2 y 2 + c 3 y 3 = c x + c 2 x 2 + c 3 x. 3
4 Question 3. [24 points] Find the general solution of the given second-order differential equation. (a) [8 points] y 7y + 2y = 0 Solution: m 2 7m + 2 = 0 = (m 3)(m 4) = 0 = m = 3,m 2 = 4 = y = c e 3x + c 2 e 4x. (b) [8 points] y 6y + 9y = 0 Solution: m 2 6m + 9 = 0 = (m 3) 2 = 0 = m = m 2 = 3 = y = c e 3x + c 2 xe 3x. (c) [8 points] y 4y + 5y = 0 Solution: m 2 4m + 5 = 0 = m,2 = 2 ± i = y = e 2x (c cos x + c 2 sin x). 4
5 Question 4. [2 points] Find the general solution of the given third-order differential equation d 3 x dt 3 + d 2 x dt 2 2x = 0 Solution: m 3 + m 2 2 = 0 = m = is one of the roots, then using Long Division, we have Then m 2,3 = ± i. Hence, the general solution is m 3 + m 2 2 = (m )(m 2 + 2m + 2) x = c e t + e t (c cos t + c 2 sin t) 5
6 Question 5. [30 points] Solve the given differential equation by undetermined coefficients, y + 2y = 2x + 5 e 2x subject to the initial conditions y(0) = 0, y (0) = 2. Solution: The associated homogeneous equation is y + 2y = 0. Then the auxiliary equation is m 2 + 2m = 0 = m = 0andm = 2. Hence, the complementary function is y c = c e 2x + c 2. Now g (x) = 2x +5 e 2x = y p (x) = Ax+B +Ce 2x, however, there is duplications between y c and y p. Then the correct assumed particular solution is y p (x) = Ax 2 + Bx+Cxe 2x. Then y p (x) = 2Ax + B +Ce 2x 2Cxe 2x and y p (x) = 2A 4Ce 2x + 4Cxe 2x. Hence, y p (x)+2y p (x) = 2A +B +4Ax 2Ce 2x = 2x +5 e 2x = 2A +2B = 5,4A = 2, 2C = = A = 2,B = 2,C = 2 = y p(x) = 2 x2 + 2x + 2 xe 2x Then the general solution is y = c e 2x + c x2 + 2x + 2 xe 2x Using I.C. = 0 = c + c 2, 2 = 2c = c =,c 2 =. Therefore, the solution is y = e 2x + 2 x2 + 2x + 2 xe 2x 6
7 Question 6. [30 points] Solve the given differential equation by variation of parameters 3y 6y + 6y = e x sec x Solution:. For 3y 6y + 6y = 0 we have 3m 2 6m + 6 = 0 3(m 2 2m + 2) = 0. So m,m 2 = 2 ± = 2 ± 2i 2 = ± i and α = = β. So,y c = e x (c cosx+c 2 sin x). Hence y = e x cos x and y 2 = e x sin x. 2. We have w(y, y 2 ) = y y 2 y y2 = e x cos x e x sin x e x cos x e x sin x e x sin x + e x cos x = e x cos x(e x sin x + e x cos x) e x sin x(e x cos x e x sin x) = e 2x cos x sin x + e 2x cos 2 x e 2x sin x cos x + e 2x sin 2 x = e 2x (cos 2 x + sin 2 x) = e 2x. 3. 3y 6y + 6y = e x sec x y 2y + 2y = 3 ex sec x. f (x) = 3 ex sec x. 4. We have So w = 0 y 2 f (x) y2 = 0 e x sin x 3 ex sec x e x sin x + e x cos x = 3 e2x sin x sec x w 2 = y 0 y f (x) = e x cos x 0 e x cos x e x sin x 3 ex sec x = 3 e2x cos x sec x. u = w w = u 2 = w 2 w = 3 e2x sin x sec x e 2x = 3 sin x sec x = 3 sin x cos x = 3 3 e2x cos x sec x e 2x = 3 cos x sec x = 3. tan x. Therefore, u = 3 tan xdx = sin x sin x 3 cos x dx = 3 cos x dx = 3 cos x d(cos x) = ln cos x. 3 u 2 = 3 dx = 3 x. 5. y p = u y + u 2 y 2 = 3 ex cos x ln cos x + 3 xex sin x. 6. y = e x (c cosx+c 2 sin x) + 3 ex cos x ln cos x + 3 xex sin x. 7
8 Question 7. [20 pionts] A 00-volt electromotive force is applied to an RC-series circuit in which the resistance is 200 ohms and the capacitance is 0 4 farad. (a) [6 points] Find the charge q(t) on the capacitor if q(0) = 0. Solution: Assume that R dq dt + C q = E(t) where R = 200 ohms, C = 0 4 farad, and E(t) = 00. Then 200 dq dt + 04 q = 00 = 2 dq dq + 00q = = dt dt + 50q = 2 = q = 00 + ce 50t Using I.C., we get c =. Hence the solution is 00 q = e 50t. (b) [4 points] Find the current i(t). Solution: i = dq dt = 00 ( 50)e 50t = 2 e 50t. 8
9 Question 8. [BONUS: 24 points] Find the general solution of x 4 y + x 3 y 4x 2 y = given that y = x 2 is a solution of the associated homogeneous equation. [HINT: Reduction of Order & Variation of Parameters] Solution: We are given that y = x 2 is a sol. of x 4 y + x 3 y 4x 2 y =. To find a second solution, we use reduction of order. Let y = x 2 u(x). Then the product rule gives y = x 2 u + 2xu and y = x 2 u + 4xu + 2n. So x 4 y + x 3 y 4x 2 y = x 5 (xu + 5u ) = 0 Letting w = u, this becomes xw + 5w = 0 Separating variables and integrating we have dw dt = 5 dx and ln w = 5ln x +C x Thus, w = x 5 and u = 4 x 4. A second solution is then y 2 = x 2 x 4 =, and the general solution of the homogeneous x 2 DE is y c = c x 2 + c 2. x 2 To find a particular solution, y p, we use variation of parameters. The Wronskian is W = 4 x. Identifying f (x) =, we obtain x 4 u = 4 x 5 = u = 6 x 4 u2 = 4 x = u 2 = 4 ln x So y p = 6 x 4 x 2 4 (ln x)x 2 = 6 x 2 4 x 2 ln x. The general solution is y = c x 2 + c 2 x 2 6 x2 ln x. 4x2 9
California State University Northridge MATH 280: Applied Differential Equations Midterm Exam 1
California State University Northridge MATH 280: Applied Differential Equations Midterm Exam 1 October 9, 2013. Duration: 75 Minutes. Instructor: Jing Li Student Name: Student number: Take your time to
More informationCalifornia State University Northridge MATH 280: Applied Differential Equations Midterm Exam 3
California State University Northridge MATH 280: Applied Differential Equations Midterm Exam 3 April 29, 2013. Duration: 75 Minutes. Instructor: Jing Li Student Name: Signature: Do not write your student
More informationSolution to Homework 2
Solution to Homework. Substitution and Nonexact Differential Equation Made Exact) [0] Solve dy dx = ey + 3e x+y, y0) = 0. Let u := e x, v = e y, and hence dy = v + 3uv) dx, du = u)dx, dv = v)dy = u)dv
More informationMAT 1341A Final Exam, 2011
MAT 1341A Final Exam, 2011 16-December, 2011. Instructor - Barry Jessup 1 Family Name: First Name: Student number: Some Advice Take 5 minutes to read the entire paper before you begin to write, and read
More information17.2 Nonhomogeneous Linear Equations. 27 September 2007
17.2 Nonhomogeneous Linear Equations 27 September 2007 Nonhomogeneous Linear Equations The differential equation to be studied is of the form ay (x) + by (x) + cy(x) = G(x) (1) where a 0, b, c are given
More informationdx n a 1(x) dy
HIGHER ORDER DIFFERENTIAL EQUATIONS Theory of linear equations Initial-value and boundary-value problem nth-order initial value problem is Solve: a n (x) dn y dx n + a n 1(x) dn 1 y dx n 1 +... + a 1(x)
More informationTest 2 - Answer Key Version A
MATH 8 Student s Printed Name: Instructor: CUID: Section: Fall 27 8., 8.2,. -.4 Instructions: You are not permitted to use a calculator on any portion of this test. You are not allowed to use any textbook,
More informationSTUDENT NAME: STUDENT SIGNATURE: STUDENT ID NUMBER: SECTION NUMBER RECITATION INSTRUCTOR:
MA262 FINAL EXAM SPRING 2016 MAY 2, 2016 TEST NUMBER 01 INSTRUCTIONS: 1. Do not open the exam booklet until you are instructed to do so. 2. Before you open the booklet fill in the information below and
More informationfor any C, including C = 0, because y = 0 is also a solution: dy
Math 3200-001 Fall 2014 Practice exam 1 solutions 2/16/2014 Each problem is worth 0 to 4 points: 4=correct, 3=small error, 2=good progress, 1=some progress 0=nothing relevant. If the result is correct,
More information3. Identify and find the general solution of each of the following first order differential equations.
Final Exam MATH 33, Sample Questions. Fall 7. y = Cx 3 3 is the general solution of a differential equation. Find the equation. Answer: y = 3y + 9 xy. y = C x + C x is the general solution of a differential
More informationMATH 2250 Final Exam Solutions
MATH 225 Final Exam Solutions Tuesday, April 29, 28, 6: 8:PM Write your name and ID number at the top of this page. Show all your work. You may refer to one double-sided sheet of notes during the exam
More information3. Identify and find the general solution of each of the following first order differential equations.
Final Exam MATH 33, Sample Questions. Fall 6. y = Cx 3 3 is the general solution of a differential equation. Find the equation. Answer: y = 3y + 9 xy. y = C x + C is the general solution of a differential
More information1 Solution to Homework 4
Solution to Homework Section. 5. The characteristic equation is r r + = (r )(r ) = 0 r = or r =. y(t) = c e t + c e t y = c e t + c e t. y(0) =, y (0) = c + c =, c + c = c =, c =. To find the maximum value
More informationReview For the Final: Problem 1 Find the general solutions of the following DEs. a) x 2 y xy y 2 = 0 solution: = 0 : homogeneous equation.
Review For the Final: Problem 1 Find the general solutions of the following DEs. a) x 2 y xy y 2 = 0 solution: y y x y2 = 0 : homogeneous equation. x2 v = y dy, y = vx, and x v + x dv dx = v + v2. dx =
More informationMA 262, Fall 2017, Final Version 01(Green)
INSTRUCTIONS MA 262, Fall 2017, Final Version 01(Green) (1) Switch off your phone upon entering the exam room. (2) Do not open the exam booklet until you are instructed to do so. (3) Before you open the
More informationUNDETERMINED COEFFICIENTS SUPERPOSITION APPROACH *
4.4 UNDETERMINED COEFFICIENTS SUPERPOSITION APPROACH 19 Discussion Problems 59. Two roots of a cubic auxiliary equation with real coeffi cients are m 1 1 and m i. What is the corresponding homogeneous
More informationDiff. Eq. App.( ) Midterm 1 Solutions
Diff. Eq. App.(110.302) Midterm 1 Solutions Johns Hopkins University February 28, 2011 Problem 1.[3 15 = 45 points] Solve the following differential equations. (Hint: Identify the types of the equations
More informationMath 201 Solutions to Assignment 1. 2ydy = x 2 dx. y = C 1 3 x3
Math 201 Solutions to Assignment 1 1. Solve the initial value problem: x 2 dx + 2y = 0, y(0) = 2. x 2 dx + 2y = 0, y(0) = 2 2y = x 2 dx y 2 = 1 3 x3 + C y = C 1 3 x3 Notice that y is not defined for some
More informationMATHEMATICS FOR ENGINEERS & SCIENTISTS 23
MATHEMATICS FOR ENGINEERS & SCIENTISTS 3.5. Second order linear O.D.E.s: non-homogeneous case.. We ll now consider non-homogeneous second order linear O.D.E.s. These are of the form a + by + c rx) for
More informationMath 116 Second Midterm November 14, 2012
Math 6 Second Midterm November 4, Name: EXAM SOLUTIONS Instructor: Section:. Do not open this exam until you are told to do so.. This exam has pages including this cover. There are 8 problems. Note that
More informationMath 106: Calculus I, Spring 2018: Midterm Exam II Monday, April Give your name, TA and section number:
Math 106: Calculus I, Spring 2018: Midterm Exam II Monday, April 6 2018 Give your name, TA and section number: Name: TA: Section number: 1. There are 6 questions for a total of 100 points. The value of
More informationLinear DifferentiaL Equation
Linear DifferentiaL Equation Massoud Malek The set F of all complex-valued functions is known to be a vector space of infinite dimension. Solutions to any linear differential equations, form a subspace
More informationMATH 3321 Sample Questions for Exam 3. 3y y, C = Perform the indicated operations, if possible: (a) AC (b) AB (c) B + AC (d) CBA
MATH 33 Sample Questions for Exam 3. Find x and y so that x 4 3 5x 3y + y = 5 5. x = 3/7, y = 49/7. Let A = 3 4, B = 3 5, C = 3 Perform the indicated operations, if possible: a AC b AB c B + AC d CBA AB
More informationSection 5.5 More Integration Formula (The Substitution Method) 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 5.5 More Integration Formula (The Substitution Method) 2 Lectures College of Science MATHS : Calculus I (University of Bahrain) Integrals / 7 The Substitution Method Idea: To replace a relatively
More informationENGI Second Order Linear ODEs Page Second Order Linear Ordinary Differential Equations
ENGI 344 - Second Order Linear ODEs age -01. Second Order Linear Ordinary Differential Equations The general second order linear ordinary differential equation is of the form d y dy x Q x y Rx dx dx Of
More informationMath 113 (Calculus 2) Exam 4
Math 3 (Calculus ) Exam 4 November 0 November, 009 Sections 0, 3 7 Name Student ID Section Instructor In some cases a series may be seen to converge or diverge for more than one reason. For such problems
More information2.3 Linear Equations 69
2.3 Linear Equations 69 2.3 Linear Equations An equation y = fx,y) is called first-order linear or a linear equation provided it can be rewritten in the special form 1) y + px)y = rx) for some functions
More informationMath 310 Introduction to Ordinary Differential Equations Final Examination August 9, Instructor: John Stockie
Make sure this exam has 15 pages. Math 310 Introduction to Ordinary Differential Equations inal Examination August 9, 2006 Instructor: John Stockie Name: (Please Print) Student Number: Special Instructions
More informationMATH 251 Examination I October 8, 2015 FORM A. Name: Student Number: Section:
MATH 251 Examination I October 8, 2015 FORM A Name: Student Number: Section: This exam has 14 questions for a total of 100 points. Show all you your work! In order to obtain full credit for partial credit
More informationMath 222 Spring 2013 Exam 3 Review Problem Answers
. (a) By the Chain ule, Math Spring 3 Exam 3 eview Problem Answers w s w x x s + w y y s (y xy)() + (xy x )( ) (( s + 4t) (s 3t)( s + 4t)) ((s 3t)( s + 4t) (s 3t) ) 8s 94st + 3t (b) By the Chain ule, w
More informationMATH 312 Section 3.1: Linear Models
MATH 312 Section 3.1: Linear Models Prof. Jonathan Duncan Walla Walla College Spring Quarter, 2007 Outline 1 Population Growth 2 Newton s Law of Cooling 3 Kepler s Law Second Law of Planetary Motion 4
More informationREFERENCE: CROFT & DAVISON CHAPTER 20 BLOCKS 1-3
IV ORDINARY DIFFERENTIAL EQUATIONS REFERENCE: CROFT & DAVISON CHAPTER 0 BLOCKS 1-3 INTRODUCTION AND TERMINOLOGY INTRODUCTION A differential equation (d.e.) e) is an equation involving an unknown function
More informationEx. 1. Find the general solution for each of the following differential equations:
MATH 261.007 Instr. K. Ciesielski Spring 2010 NAME (print): SAMPLE TEST # 2 Solve the following exercises. Show your work. (No credit will be given for an answer with no supporting work shown.) Ex. 1.
More informationAPPLIED MATHEMATICS. Part 1: Ordinary Differential Equations. Wu-ting Tsai
APPLIED MATHEMATICS Part 1: Ordinary Differential Equations Contents 1 First Order Differential Equations 3 1.1 Basic Concepts and Ideas................... 4 1.2 Separable Differential Equations................
More informationOrdinary Differential Equations Lake Ritter, Kennesaw State University
Ordinary Differential Equations Lake Ritter, Kennesaw State University 2017 MATH 2306: Ordinary Differential Equations Lake Ritter, Kennesaw State University This manuscript is a text-like version of the
More information0.1 Problems to solve
0.1 Problems to solve Homework Set No. NEEP 547 Due September 0, 013 DLH Nonlinear Eqs. reducible to first order: 1. 5pts) Find the general solution to the differential equation: y = [ 1 + y ) ] 3/. 5pts)
More informationLesson 3: Linear differential equations of the first order Solve each of the following differential equations by two methods.
Lesson 3: Linear differential equations of the first der Solve each of the following differential equations by two methods. Exercise 3.1. Solution. Method 1. It is clear that y + y = 3 e dx = e x is an
More informationCalifornia State University Northridge MATH 255A: Calculus for the Life Sciences I Midterm Exam 3
California State University Northrige MATH 255A: Calculus for the Life Sciences I Mierm Exam 3 Due May 8 2013. Instructor: Jing Li Stuent Name: Signature: Do not write your stuent ID number on this front
More informationMATH 23 Exam 2 Review Solutions
MATH 23 Exam 2 Review Solutions Problem 1. Use the method of reduction of order to find a second solution of the given differential equation x 2 y (x 0.1875)y = 0, x > 0, y 1 (x) = x 1/4 e 2 x Solution
More information2. Determine whether the following pair of functions are linearly dependent, or linearly independent:
Topics to be covered on the exam include: Recognizing, and verifying solutions to homogeneous second-order linear differential equations, and their corresponding Initial Value Problems Recognizing and
More informationPhys 2025, First Test. September 20, minutes Name:
Phys 05, First Test. September 0, 011 50 minutes Name: Show all work for maximum credit. Each problem is worth 10 points. Work 10 of the 11 problems. k = 9.0 x 10 9 N m / C ε 0 = 8.85 x 10-1 C / N m e
More information- - - - - - - - - - - - - - - - - - DISCLAIMER - - - - - - - - - - - - - - - - - - General Information: This midterm is a sample midterm. This means: The sample midterm contains problems that are of similar,
More informationMath 2142 Homework 5 Part 1 Solutions
Math 2142 Homework 5 Part 1 Solutions Problem 1. For the following homogeneous second order differential equations, give the general solution and the particular solution satisfying the given initial conditions.
More informationMATH 101: PRACTICE MIDTERM 2
MATH : PRACTICE MIDTERM INSTRUCTOR: PROF. DRAGOS GHIOCA March 7, Duration of examination: 7 minutes This examination includes pages and 6 questions. You are responsible for ensuring that your copy of the
More informationAPPM 2360 Section Exam 3 Wednesday November 19, 7:00pm 8:30pm, 2014
APPM 2360 Section Exam 3 Wednesday November 9, 7:00pm 8:30pm, 204 ON THE FRONT OF YOUR BLUEBOOK write: () your name, (2) your student ID number, (3) lecture section, (4) your instructor s name, and (5)
More information5.4 Variation of Parameters
202 5.4 Variation of Parameters The method of variation of parameters applies to solve (1) a(x)y + b(x)y + c(x)y = f(x). Continuity of a, b, c and f is assumed, plus a(x) 0. The method is important because
More informationTHE USE OF CALCULATORS, BOOKS, NOTES ETC. DURING THIS EXAMINATION IS PROHIBITED. Do not write in the blanks below. 1. (5) 7. (12) 2. (5) 8.
MATH 4 EXAMINATION II MARCH 24, 2004 TEST FORM A NAME STUDENT NUMBER INSTRUCTOR SECTION NUMBER This examination consists of 2 problems. The first 6 are multiple choice questions, the next two are short
More information- - - - - - - - - - - - - - - - - - DISCLAIMER - - - - - - - - - - - - - - - - - - General Information: This is a midterm from a previous semester. This means: This midterm contains problems that are of
More informationMath 266 Midterm Exam 2
Math 266 Midterm Exam 2 March 2st 26 Name: Ground Rules. Calculator is NOT allowed. 2. Show your work for every problem unless otherwise stated (partial credits are available). 3. You may use one 4-by-6
More informationMath 116 Second Midterm November 17, 2010
Math 6 Second Midterm November 7, Name: EXAM SOLUTIONS Instructor: Section:. Do not open this exam until you are told to do so.. This exam has pages including this cover. There are problems. Note that
More informationDO NOT BEGIN THIS TEST UNTIL INSTRUCTED TO START
Math 265 Student name: KEY Final Exam Fall 23 Instructor & Section: This test is closed book and closed notes. A (graphing) calculator is allowed for this test but cannot also be a communication device
More information- - - - - - - - - - - - - - - - - - DISCLAIMER - - - - - - - - - - - - - - - - - - General Information: This midterm is a sample midterm. This means: The sample midterm contains problems that are of similar,
More informationStudent s Printed Name:
Student s Printed Name: Instructor: CUID: Section # : You are not permitted to use a calculator on any part of this test. You are not allowed to use any textbook, notes, cell phone, laptop, PDA, or any
More informationMath 122 Fall Unit Test 1 Review Problems Set A
Math Fall 8 Unit Test Review Problems Set A We have chosen these problems because we think that they are representative of many of the mathematical concepts that we have studied. There is no guarantee
More informationSection 3.4. Second Order Nonhomogeneous. The corresponding homogeneous equation. is called the reduced equation of (N).
Section 3.4. Second Order Nonhomogeneous Equations y + p(x)y + q(x)y = f(x) (N) The corresponding homogeneous equation y + p(x)y + q(x)y = 0 (H) is called the reduced equation of (N). 1 General Results
More informationSTUDENT NAME: STUDENT SIGNATURE: STUDENT ID NUMBER: SECTION NUMBER RECITATION INSTRUCTOR:
MA262 EXAM I SPRING 2016 FEBRUARY 25, 2016 TEST NUMBER 01 INSTRUCTIONS: 1. Do not open the exam booklet until you are instructed to do so. 2. Before you open the booklet fill in the information below and
More informationMath Exam 2, October 14, 2008
Math 96 - Exam 2, October 4, 28 Name: Problem (5 points Find all solutions to the following system of linear equations, check your work: x + x 2 x 3 2x 2 2x 3 2 x x 2 + x 3 2 Solution Let s perform Gaussian
More information20D - Homework Assignment 4
Brian Bowers (TA for Hui Sun) MATH 0D Homework Assignment November, 03 0D - Homework Assignment First, I will give a brief overview of how to use variation of parameters. () Ensure that the differential
More informationProblem Max. Possible Points Total
MA 262 Exam 1 Fall 2011 Instructor: Raphael Hora Name: Max Possible Student ID#: 1234567890 1. No books or notes are allowed. 2. You CAN NOT USE calculators or any electronic devices. 3. Show all work
More informationPRELIMINARY THEORY LINEAR EQUATIONS
4.1 PRELIMINARY THEORY LINEAR EQUATIONS 117 4.1 PRELIMINARY THEORY LINEAR EQUATIONS REVIEW MATERIAL Reread the Remarks at the end of Section 1.1 Section 2.3 (especially page 57) INTRODUCTION In Chapter
More informationHomework Solutions: , plus Substitutions
Homework Solutions: 2.-2.2, plus Substitutions Section 2. I have not included any drawings/direction fields. We can see them using Maple or by hand, so we ll be focusing on getting the analytic solutions
More informationMAT292 - Calculus III - Fall Solution for Term Test 2 - November 6, 2014 DO NOT WRITE ON THE QR CODE AT THE TOP OF THE PAGES.
MAT9 - Calculus III - Fall 4 Solution for Term Test - November 6, 4 Time allotted: 9 minutes. Aids permitted: None. Full Name: Last First Student ID: Email: @mail.utoronto.ca Instructions DO NOT WRITE
More informationProblem # Max points possible Actual score Total 100
MIDTERM 1-18.01 - FALL 2014. Name: Email: Please put a check by your recitation section. Instructor Time B.Yang MW 10 M. Hoyois MW 11 M. Hoyois MW 12 X. Sun MW 1 R. Chang MW 2 Problem # Max points possible
More informationx gm 250 L 25 L min y gm min
Name NetID MATH 308 Exam 2 Spring 2009 Section 511 Hand Computations P. Yasskin Solutions 1 /10 4 /30 2 /10 5 /30 3 /10 6 /15 Total /105 1. (10 points) Tank X initially contains 250 L of sugar water with
More informationSecond-Order Linear ODEs
C0.tex /4/011 16: 3 Page 13 Chap. Second-Order Linear ODEs Chapter presents different types of second-order ODEs and the specific techniques on how to solve them. The methods are systematic, but it requires
More informationSection 3.4. Second Order Nonhomogeneous. The corresponding homogeneous equation
Section 3.4. Second Order Nonhomogeneous Equations y + p(x)y + q(x)y = f(x) (N) The corresponding homogeneous equation y + p(x)y + q(x)y = 0 (H) is called the reduced equation of (N). 1 General Results
More informationMath 152 Take Home Test 1
Math 5 Take Home Test Due Monday 5 th October (5 points) The following test will be done at home in order to ensure that it is a fair and representative reflection of your own ability in mathematics I
More informationSecond-Order Linear ODEs
Chap. 2 Second-Order Linear ODEs Sec. 2.1 Homogeneous Linear ODEs of Second Order On pp. 45-46 we extend concepts defined in Chap. 1, notably solution and homogeneous and nonhomogeneous, to second-order
More informationIntegration by Substitution
November 22, 2013 Introduction 7x 2 cos(3x 3 )dx =? 2xe x2 +5 dx =? Chain rule The chain rule: d dx (f (g(x))) = f (g(x)) g (x). Use the chain rule to find f (x) and then write the corresponding anti-differentiation
More informationChapter 3. Reading assignment: In this chapter we will cover Sections dx 1 + a 0(x)y(x) = g(x). (1)
Chapter 3 3 Introduction Reading assignment: In this chapter we will cover Sections 3.1 3.6. 3.1 Theory of Linear Equations Recall that an nth order Linear ODE is an equation that can be written in the
More informationMA 262, Spring 2018, Midterm 1 Version 01 (Green)
MA 262, Spring 2018, Midterm 1 Version 01 (Green) INSTRUCTIONS 1. Switch off your phone upon entering the exam room. 2. Do not open the exam booklet until you are instructed to do so. 3. Before you open
More informationMath 116 Second Midterm November 16, 2011
Math 6 Second Midterm November 6, Name: EXAM SOLUTIONS Instructor: Section:. Do not open this exam until you are told to do so.. This exam has pages including this cover. There are 9 problems. Note that
More informationTest 3 - Answer Key Version B
Student s Printed Name: Instructor: CUID: Section: Instructions: You are not permitted to use a calculator on any portion of this test. You are not allowed to use any textbook, notes, cell phone, laptop,
More informationMAE140 - Linear Circuits - Winter 09 Midterm, February 5
Instructions MAE40 - Linear ircuits - Winter 09 Midterm, February 5 (i) This exam is open book. You may use whatever written materials you choose, including your class notes and textbook. You may use a
More informationPrelim 1 Solutions V2 Math 1120
Feb., Prelim Solutions V Math Please show your reasoning and all your work. This is a 9 minute exam. Calculators are not needed or permitted. Good luck! Problem ) ( Points) Calculate the following: x a)
More informationSpring 2016 Exam 1 without number 13.
MARK BOX problem points 0 5-9 45 without number 3. (Topic of number 3 is not on our Exam this semester.) Solutions on homepage (under previous exams). 0 0 0 NAME: 2 0 3 0 PIN: % 00 INSTRUCTIONS On Problem
More informationTHE USE OF A CALCULATOR, CELL PHONE, OR ANY OTHER ELECTRONIC DEVICE IS NOT PERMITTED IN THIS EXAMINATION.
MATH 110 FINAL EXAM SPRING 2008 FORM A NAME STUDENT NUMBER INSTRUCTOR SECTION NUMBER This examination will be machine processed by the University Testing Service. Use only a number 2 pencil on your scantron.
More informationEXAM. Exam #1. Math 3350 Summer II, July 21, 2000 ANSWERS
EXAM Exam #1 Math 3350 Summer II, 2000 July 21, 2000 ANSWERS i 100 pts. Problem 1. 1. In each part, find the general solution of the differential equation. dx = x2 e y We use the following sequence of
More informationكلية العلوم قسم الرياضيات المعادالت التفاضلية العادية
الجامعة اإلسالمية كلية العلوم غزة قسم الرياضيات المعادالت التفاضلية العادية Elementary differential equations and boundary value problems المحاضرون أ.د. رائد صالحة د. فاتن أبو شوقة 1 3 4 5 6 بسم هللا
More informationReview for Exam 2. Review for Exam 2.
Review for Exam 2. 5 or 6 problems. No multiple choice questions. No notes, no books, no calculators. Problems similar to homeworks. Exam covers: Regular-singular points (5.5). Euler differential equation
More informationPolytechnic Institute of NYU MA 2132 Final Practice Answers Fall 2012
Polytechnic Institute of NYU MA Final Practice Answers Fall Studying from past or sample exams is NOT recommended. If you do, it should be only AFTER you know how to do all of the homework and worksheet
More informationECONOMICS 207 SPRING 2006 LABORATORY EXERCISE 5 KEY. 8 = 10(5x 2) = 9(3x + 8), x 50x 20 = 27x x = 92 x = 4. 8x 2 22x + 15 = 0 (2x 3)(4x 5) = 0
ECONOMICS 07 SPRING 006 LABORATORY EXERCISE 5 KEY Problem. Solve the following equations for x. a 5x 3x + 8 = 9 0 5x 3x + 8 9 8 = 0(5x ) = 9(3x + 8), x 0 3 50x 0 = 7x + 7 3x = 9 x = 4 b 8x x + 5 = 0 8x
More information= 2e t e 2t + ( e 2t )e 3t = 2e t e t = e t. Math 20D Final Review
Math D Final Review. Solve the differential equation in two ways, first using variation of parameters and then using undetermined coefficients: Corresponding homogenous equation: with characteristic equation
More information- - - - - - - - - - - - - - - - - - DISCLAIMER - - - - - - - - - - - - - - - - - - General Information: This midterm is a sample midterm. This means: The sample midterm contains problems that are of similar,
More informationMath 116 Practice for Exam 2
Math 6 Practice for Exam Generated October 6, 5 Name: Instructor: Section Number:. This exam has 5 questions. Note that the problems are not of equal difficulty, so you may want to skip over and return
More informationMath 240 Calculus III
Calculus III Summer 2015, Session II Monday, August 3, 2015 Agenda 1. 2. Introduction The reduction of technique, which applies to second- linear differential equations, allows us to go beyond equations
More informationMath 308 Week 8 Solutions
Math 38 Week 8 Solutions There is a solution manual to Chapter 4 online: www.pearsoncustom.com/tamu math/. This online solutions manual contains solutions to some of the suggested problems. Here are solutions
More informationFACULTY OF ARTS AND SCIENCE University of Toronto FINAL EXAMINATIONS, APRIL 2016 MAT 133Y1Y Calculus and Linear Algebra for Commerce
FACULTY OF ARTS AND SCIENCE University of Toronto FINAL EXAMINATIONS, APRIL 206 MAT YY Calculus and Linear Algebra for Commerce Duration: Examiners: hours N. Hoell A. Igelfeld D. Reiss L. Shorser J. Tate
More informationMixing Problems. Solution of concentration c 1 grams/liter flows in at a rate of r 1 liters/minute. Figure 1.7.1: A mixing problem.
page 57 1.7 Modeling Problems Using First-Order Linear Differential Equations 57 For Problems 33 38, use a differential equation solver to determine the solution to each of the initial-value problems and
More informationNovember 20, Problem Number of points Points obtained Total 50
MATH 124 E MIDTERM 2, v.b Autumn 2018 November 20, 2018 NAME: SIGNATURE: STUDENT ID #: GAB AB AB AB AB AB AB AB AB AB AB AB AB AB QUIZ SECTION: ABB ABB Problem Number of points Points obtained 1 14 2 10
More informationAPPM 2360: Final Exam 10:30am 1:00pm, May 6, 2015.
APPM 23: Final Exam :3am :pm, May, 25. ON THE FRONT OF YOUR BLUEBOOK write: ) your name, 2) your student ID number, 3) lecture section, 4) your instructor s name, and 5) a grading table for eight questions.
More informationMath 217 Fall 2000 Exam Suppose that y( x ) is a solution to the differential equation
Math 17 Fall 000 Exam 1 Notational Remark: In this exam, the symbol x y( x ) means dy dx. 1. Suppose that y( x ) is a solution to the differential equation, x y( x ) F ( x, y( x )) y( x ) 0 y 0. Then y'(x
More informationMath 322. Spring 2015 Review Problems for Midterm 2
Linear Algebra: Topic: Linear Independence of vectors. Question. Math 3. Spring Review Problems for Midterm Explain why if A is not square, then either the row vectors or the column vectors of A are linearly
More informationReview Problems for Exam 2
Review Problems for Exam 2 This is a list of problems to help you review the material which will be covered in the final. Go over the problem carefully. Keep in mind that I am going to put some problems
More informationTurn off all cell phones, pagers, radios, mp3 players, and other similar devices.
Math 25 B and C Midterm 2 Palmieri, Autumn 26 Your Name Your Signature Student ID # TA s Name and quiz section (circle): Cady Cruz Jacobs BA CB BB BC CA CC Turn off all cell phones, pagers, radios, mp3
More informationStudent name: Student ID: TA s name and/or section: MATH 3B (Butler) Midterm II, 20 February 2009
Student name: Student ID: TA s name and/or section: MATH 3B (Butler) Midterm II, 20 February 2009 This test is closed book and closed notes. No calculator is allowed for this test. For full credit show
More informationMath 2Z03 - Tutorial # 6. Oct. 26th, 27th, 28th, 2015
Math 2Z03 - Tutorial # 6 Oct. 26th, 27th, 28th, 2015 Tutorial Info: Tutorial Website: http://ms.mcmaster.ca/ dedieula/2z03.html Office Hours: Mondays 3pm - 5pm (in the Math Help Centre) Tutorial #6: 3.4
More informationYORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics. MATH A Test #2. June 25, 2014 SOLUTIONS
YORK UNIVERSITY Faculty of Science Department of Mathematics an Statistics MATH 505 6.00 A Test # June 5, 04 SOLUTIONS Family Name (print): Given Name: Stuent No: Signature: INSTRUCTIONS:. Please write
More informationTest 2 - Answer Key Version A
MATH 8 Student s Printed Name: Instructor: Test - Answer Key Spring 6 8. - 8.3,. -. CUID: Section: Instructions: You are not permitted to use a calculator on any portion of this test. You are not allowed
More informationFinal exam (practice) UCLA: Math 31B, Spring 2017
Instructor: Noah White Date: Final exam (practice) UCLA: Math 3B, Spring 207 This exam has 8 questions, for a total of 80 points. Please print your working and answers neatly. Write your solutions in the
More information