Section 4.3 version November 3, 2011 at 23:18 Exercises 2. The equation to be solved is
|
|
- Jesse Booker
- 5 years ago
- Views:
Transcription
1 Section 4.3 version November 3, 011 at 3:18 Exercises. The equation to be solved is y (t)+y (t)+6y(t) = 0. The characteristic equation is λ +λ+6 = 0. The solutions are λ 1 = and λ = 3. Therefore, y 1 (t) = e t and y (t) = e 3t are solutions of the given equation. Now calculate the Wronskian W ( e t,e 3t) e t e 3t = e t 3e 3t = e t ( 3)e 3t e 3t ( )e t = e t 0. Nonzero Wronskian implies that these solutions are linearly independent. Thus they form a fundamental set of solutions. The general solution is y(t) = C 1 e t +C e 3t. 9. The equation to be solved is y (t)+4y (t)+y(t) = 0. The characteristic equation is λ +4λ+ = 0. The solutions are λ 1 = +i and λ == i. Therefore, y 1 (t) = e t cost and y (t) = e t sint are solutions of the given equation. The Wronskian is not zero, so these solutions are linearly independent. Thus they form a fundamental set of solutions. The general solution is y(t) = C 1 e t cost+c e t sint = e t( C 1 cost+c sint ) = Ae t cos(t φ). 1. The initial value problem to be solved is y (t)+y(t) = 0, y(0) = 1, y (0) = 1.
2 First we find the general solution. The characteristic equation is λ + = 0. The solutions are λ 1 = i and λ = i. Therefore, y 1 (t) = e it and y (t) = e it are solutions of the given equation. But, we are interested in real solutions. So, use Euler s formula to get the real and imaginary part: y 1 = e it = cos(t)+isin(t) and y = e it = cos(t) isin(t). As we showed in class the real part and the imaginary part of these functions are also solutions of the given equation. Now calculate the Wronskian W ( cos(t),sin(t) ) cos(t) sin(t) = sin(t) cos(t) = (cos(t)) +(sin(t)) = 0. Nonzero Wronskian implies that cos(t) and sin(t) are linearly independent. Thus they form a fundamental set of solutions of the given equation. The general solution is y(t) = C 1 cos(t)+c sin(t). To solve the initial value problem we find the derivative first: y (t) = C 1 sin(t)+c cos(t). Now we use the initial conditions: 1 = y(0) = C 1 1+C 0, 1 = y (0) = C 1 0+C 1. Thus C 1 = 1 and C = 1. The solution of the given initial value problem is y(t) = cos(t) 1 sin(t). But, much more preferable way to write the solution is in the form A cos(t φ). To do this we use complex numbers: cos(t) = Re ( e it), 1 ( ) i sin(t) = Re eit
3 Thus cos(t) 1 ( sin(t) = Re e it + i ) (( eit = Re 1+ i ) )e it. Now we convert 1+i/ to polar form 1+ i = 1+ 1 eiθ = 6 where θ = arctan ( 1/ ). Hence, ( 1+ i ) 6 6 e it = e iθ e it = e i(t+θ), and therefore cos(t) 1 (( sin(t) = Re 1+ i ) )e it = Re ( 6 e iθ, e i(t+θ) ) = 6 cos( t+θ ). Thus the solution is y(t) = cos(t) 1 sin(t) = 6 cos( t φ ), where φ = arctan ( 1/ ). Just to make sure that the last two formulas represent the same function I plot them on the same graph, see Figure 1.. The initial value problem to be solved is y (t)+10y (t)+y(t) = 0, y(0) =, y (0) = 1. First we find the general solution. The characteristic equation is λ +10λ+ = 0. There is only one solution λ 1 =. The corresponding solution of the given differential equation is y 1 (t) = e t. As it is explained in the book the other solution is The solutions y (t) = te t. y 1 (t) = e t and y (t) = te t are linearly independent since their Wronskian is nonzero: W ( e t,te t) e t te t = e t e t te t = e 10t te 10t +te 10t = e 10t 0.
4 Π 10 Π 3Π 10 Π Π 3Π 7Π 10 4Π 9Π 10 Π Figure 1: Problem 1 The general solution is y(t) = C 1 e t +C te t. To solve the initial value problem we find the derivative first: y (t) = C 1 e t +C e t C te t = ( C 1 +C C t ) e t. Now we use the initial conditions: = y(0) = C 1 1+C 0, 1 = y (0) = C 1 +C. Thus C 1 = and C = 9. The solution of the given initial value problem is, see Figure, y(t) = e t +9te t = ( +9t ) e t. My comment. The initial value problem that we solved here can be interpreted as describing the motion of a mass attached to a spring with damping. Here the mass is 1kg, damping is 10kg/s and spring constant kg/s. At time 0 the mass is displaced by m from the equilibrium position and it is given the initial velocity -1m/s. The solution that we found
5 Figure : Problem describes the position of the mass at any time t > 0. It means that the mass will just creep back to the equilibrium position. An interesting question that arises here is whether we can give the mass an initial velocity so that it passes through the equilibrium. To solve this problem we need to set the initial velocity as an unknown quantity, call it v 0 and solve the initial value problem: y (t)+10y (t)+y(t) = 0, y(0) =, y (0) = v 0. The solution of this problem is y(t) = ( +(10+v 0 )t ) e t. Now the question is: for which v 0 there exists t > 0 such that y(t) = 0. Since e t > 0, we need a positive t such that +(10+v 0 )t = 0. Solving for t we get t = /(10 + v 0 ). This quantity will be positive if v 0 < 10. For example, for v 0 = 18 we have t = 1/4 see Figure 3. Looking at Figure 3, a natural question arises: What is the lowest position that the mass will reach in this situation? Here we assume that v 0 = 18 and the solution is y(t) = ( 8t ) e t.
6 Figure 3: Problem, my comments To answer the question, we take the derivative y (t) = ( 8t ) e t 8e t = ( 18+40t ) e t. Solving y (t) = 0 for t > 0 we get t = 9/0. The lowest position of the mass is y(9/0) = ( 89/0 ) e 9/0 = (8/)e 9/ The initial value problem to be solved is y (t) 4y (t) y(t) = 0, y(1) = 1, y (1) = 1. First we find the general solution. The characteristic equation is λ 4λ = 0. The solutions are λ 1 = 1 and λ =. Therefore, y 1 (t) = e t and y (t) = e t are solutions of the given equation. These solutions are linearly independent. The general solution is y(t) = C 1 e t +C e t.
7 To solve the initial value problem we find the derivative first: Now we use the initial conditions: y (t) = C 1 e t +C e t. 1 = y(1) = C 1 e 1 +C e, 1 = y (1) = C 1 e 1 +C e. Thus we need to solve C 1 e 1 +C e = 1 C 1 e 1 +C e = 1 We can always solve one equation and substitute the solution into the other equation. But, here it is easier to add two equations to get 6C e =. Thus C = e /3. Substituting this in the first equation we get C 1 e 1 1/3 = 1. Thus C 1 = e/3. Finally, the solution is y(t) = e 3 e t e 3 et = 1 ) (e 1 t +e (t 1) The initial value problem to be solved is 4y (t)+y(t) = 0, y(1) = 0, y (1) =. First we find the general solution. The characteristic equation is 4λ +1 = 0. The solutions are λ 1 = i/ and λ = i/. The complex solutions are e it/ = cos ( t/ ) +isin ( t/ ), e it/ = cos ( t/ ) isin ( t/ ). The real solutions are cos ( t/ ), sin ( t/ ) These solutions are linearly independent. The general solution is y(t) = C 1 cos ( t/ ) +C sin ( t/ ). To solve the initial value problem we find the derivative first: y (t) = C 1 sin( t/ ) + C cos( t/ ).
8 Now we use the initial conditions: 0 = y(1) = C 1 cos(1/)+c sin(1/), = y (1) = C 1 sin( 1/ ) + C cos( 1/ ). Thus we need to solve C 1 cos(1/)+c sin(1/) = 0 C 1 sin( 1/ ) + C cos( 1/ ) = Solve the first equation for C 1 and substitute the solution into the second equation: C 1 = C sin(1/)/cos(1/), C 1 Solve the last equation for C : C 1 (sin(1/)) +(cos(1/)) cos(1/) sin(1/) cos(1/) sin(1/)+c 1 cos(1/) =. 1 1 C cos(1/) = C = 4cos(1/) = Now sin(1/) sin(1/) C 1 = C = 4 cos(1/) cos(1/) cos(1/) = 4sin(1/). Finally, the solution is (see Figure 4) y(t) = 4sin(1/)cos ( t/ ) 4cos(1/)sin ( t/ ). My comment. It is interesting to determine the amplitude and the phase of this solution. ) y(t) = Re (4sin(1/)e it/ +4cos(1/)ie it/ ( (sin(1/)+i ) = 4Re cos(1/) e it/) = 4Re (i ( cos(1/) isin(1/) ) ) e it/ ) = 4Re (ie i1/ e it/ ) = 4Re (e iπ/ e i1/ e it/ = 4Re ( e i(t 1+π)/) = 4cos ( (t 1+π)/ ) Thus the amplitude and the phase are A = 4 and φ = 1 π.
9 Π 1 Π Π 1 Π 1 3Π 1 4Π 1 Π 1 6Π 1 7Π 3 4 Figure 4: Problem 6
Solutions for homework 5
1 Section 4.3 Solutions for homework 5 17. The following equation has repeated, real, characteristic roots. Find the general solution. y 4y + 4y = 0. The characteristic equation is λ 4λ + 4 = 0 which has
More informationDifferential Equations: Homework 8
Differential Equations: Homework 8 Alvin Lin January 08 - May 08 Section.6 Exercise Find a general solution to the differential equation using the method of variation of parameters. y + y = tan(t) r +
More information3.4 Application-Spring Mass Systems (Unforced and frictionless systems)
3.4. APPLICATION-SPRING MASS SYSTEMS (UNFORCED AND FRICTIONLESS SYSTEMS)73 3.4 Application-Spring Mass Systems (Unforced and frictionless systems) Second order differential equations arise naturally when
More informationSolutions to the Homework Replaces Section 3.7, 3.8
Solutions to the Homework Replaces Section 3.7, 3.8 1. Our text (p. 198) states that µ ω 0 = ( 1 γ2 4km ) 1/2 1 1 2 γ 2 4km How was this approximation made? (Hint: Linearize 1 x) SOLUTION: We linearize
More informationLecture Notes for Math 251: ODE and PDE. Lecture 12: 3.3 Complex Roots of the Characteristic Equation
Lecture Notes for Math 21: ODE and PDE. Lecture 12: 3.3 Complex Roots of the Characteristic Equation Shawn D. Ryan Spring 2012 1 Complex Roots of the Characteristic Equation Last Time: We considered the
More informationMA2327, Problem set #3 (Practice problems with solutions)
MA2327, Problem set #3 (Practice problems with solutions) 5 2 Compute the matrix exponential e ta in the case that A = 2 5 2 Compute the matrix exponential e ta in the case that A = 5 3 Find the unique
More informationEven-Numbered Homework Solutions
-6 Even-Numbered Homework Solutions Suppose that the matric B has λ = + 5i as an eigenvalue with eigenvector Y 0 = solution to dy = BY Using Euler s formula, we can write the complex-valued solution Y
More informationWork sheet / Things to know. Chapter 3
MATH 251 Work sheet / Things to know 1. Second order linear differential equation Standard form: Chapter 3 What makes it homogeneous? We will, for the most part, work with equations with constant coefficients
More informationMATH 251 Week 6 Not collected, however you are encouraged to approach all problems to prepare for exam
MATH 51 Week 6 Not collected, however you are encouraged to approach all problems to prepare for exam A collection of previous exams could be found at the coordinator s web: http://www.math.psu.edu/tseng/class/m51samples.html
More informationAPPM 2360: Midterm 3 July 12, 2013.
APPM 2360: Midterm 3 July 12, 2013. ON THE FRONT OF YOUR BLUEBOOK write: (1) your name, (2) your instructor s name, (3) your recitation section number and (4) a grading table. Text books, class notes,
More informationMath53: Ordinary Differential Equations Autumn 2004
Math53: Ordinary Differential Equations Autumn 2004 Unit 2 Summary Second- and Higher-Order Ordinary Differential Equations Extremely Important: Euler s formula Very Important: finding solutions to linear
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 informationExaminer: D. Burbulla. Aids permitted: Formula Sheet, and Casio FX-991 or Sharp EL-520 calculator.
University of Toronto Faculty of Applied Science and Engineering Solutions to Final Examination, June 017 Duration: and 1/ hrs First Year - CHE, CIV, CPE, ELE, ENG, IND, LME, MEC, MMS MAT187H1F - Calculus
More informationComplex Numbers and Exponentials
Complex Numbers and Exponentials Definition and Basic Operations A complexnumber is nothing morethan a point in the xy plane. The first component, x, ofthe complex number (x,y) is called its real part
More informationc 1 = y 0, c 2 = 1 2 y 1. Therefore the solution to the general initial-value problem is y(t) = y 0 cos(2t)+y sin(2t).
Solutions to Second In-Class Exam Math 246, Professor David Levermore Tuesday, 29 October 2 ( [4] Give the interval of definition for the solution of the initial-value problem u t u + cos(5t 6+t u = et
More informationSection 7.1 Exercises
Section 7. Solving Trigonometric Equations and Identities 5 Section 7. Eercises Find all solutions on the interval sin. sin.. cos. cos Find all solutions 5. sin 9. cos 5 6. sin. 8cos 6 7. cos t 8. cos
More informationMATH 251 Examination I October 5, 2017 FORM A. Name: Student Number: Section:
MATH 251 Examination I October 5, 2017 FORM A Name: Student Number: Section: This exam has 13 questions for a total of 100 points. Show all your work! In order to obtain full credit for partial credit
More informationSection 7.1 Exercises
Section 7.1 Solving Trigonometric Equations and Identities 109 Section 7.1 Eercises Find all solutions on the interval 0 sin 1. sin 1.. cos 1. cos Find all solutions 5. sin 1 9. cos 5 6. sin 10. 8cos 6
More informationDON T PANIC! If you get stuck, take a deep breath and go on to the next question. Come back to the question you left if you have time at the end.
Math 307, Midterm 2 Winter 2013 Name: Instructions. DON T PANIC! If you get stuck, take a deep breath and go on to the next question. Come back to the question you left if you have time at the end. There
More informationMATH 251 Examination I October 9, 2014 FORM A. Name: Student Number: Section:
MATH 251 Examination I October 9, 2014 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 informationSection 4.9; Section 5.6. June 30, Free Mechanical Vibrations/Couple Mass-Spring System
Section 4.9; Section 5.6 Free Mechanical Vibrations/Couple Mass-Spring System June 30, 2009 Today s Session Today s Session A Summary of This Session: Today s Session A Summary of This Session: (1) Free
More informationMath K (24564) - Lectures 02
Math 39100 K (24564) - Lectures 02 Ethan Akin Office: NAC 6/287 Phone: 650-5136 Email: ethanakin@earthlink.net Spring, 2018 Contents Second Order Linear Equations, B & D Chapter 4 Second Order Linear Homogeneous
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 informationExam 3 Review Sheet Math 2070
The syllabus for Exam 3 is Sections 3.6, 5.1 to 5.3, 5.6, and 6.1 to 6.4. You should review the assigned exercises in these sections. Following is a brief list (not necessarily complete of terms, skills,
More information4.2 Homogeneous Linear Equations
4.2 Homogeneous Linear Equations Homogeneous Linear Equations with Constant Coefficients Consider the first-order linear differential equation with constant coefficients a 0 and b. If f(t) = 0 then this
More informationMath 308 Exam II Practice Problems
Math 38 Exam II Practice Problems This review should not be used as your sole source for preparation for the exam. You should also re-work all examples given in lecture and all suggested homework problems..
More informationMATH 246: Chapter 2 Section 8 Motion Justin Wyss-Gallifent
MATH 46: Chapter Section 8 Motion Justin Wyss-Gallifent 1. Introduction Important: Positive is up and negative is down. Imagine a spring hanging with no weight on it. We then attach a mass m which stretches
More informationCalculus IV - HW 3. Due 7/ Give the general solution to the following differential equations: y = c 1 e 5t + c 2 e 5t. y = c 1 e 2t + c 2 e 4t.
Calculus IV - HW 3 Due 7/13 Section 3.1 1. Give the general solution to the following differential equations: a y 25y = 0 Solution: The characteristic equation is r 2 25 = r 5r + 5. It follows that the
More informationGraded and supplementary homework, Math 2584, Section 4, Fall 2017
Graded and supplementary homework, Math 2584, Section 4, Fall 2017 (AB 1) (a) Is y = cos(2x) a solution to the differential equation d2 y + 4y = 0? dx2 (b) Is y = e 2x a solution to the differential equation
More informationMATH 251 Examination I July 5, 2011 FORM A. Name: Student Number: Section:
MATH 251 Examination I July 5, 2011 FORM A Name: Student Number: Section: This exam has 12 questions for a total of 100 points. Show all you your work! In order to obtain full credit for partial credit
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 informationMATH 251 Examination I February 23, 2017 FORM A. Name: Student Number: Section:
MATH 251 Examination I February 23, 2017 FORM A Name: Student Number: Section: This exam has 12 questions for a total of 100 points. Show all you your work! In order to obtain full credit for partial credit
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 informationSolutions to the Homework Replaces Section 3.7, 3.8
Solutions to the Homework Replaces Section 3.7, 3.8. Show that the period of motion of an undamped vibration of a mass hanging from a vertical spring is 2π L/g SOLUTION: With no damping, mu + ku = 0 has
More information1. (10 points) Find the general solution to the following second-order differential equation:
Math 307A, Winter 014 Midterm Solutions Page 1 of 8 1. (10 points) Find the general solution to the following second-order differential equation: 4y 1y + 9y = 9t. To find the general solution to this nonhomogeneous
More informationMATH 307 Introduction to Differential Equations Autumn 2017 Midterm Exam Monday November
MATH 307 Introduction to Differential Equations Autumn 2017 Midterm Exam Monday November 6 2017 Name: Student ID Number: I understand it is against the rules to cheat or engage in other academic misconduct
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 informationMath K (24564) - Homework Solutions 02
Math 39100 K (24564) - Homework Solutions 02 Ethan Akin Office: NAC 6/287 Phone: 650-5136 Email: ethanakin@earthlink.net Spring, 2018 Contents Reduction of Order, B & D Chapter 3, p. 174 Constant Coefficient
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 information4.9 Free Mechanical Vibrations
4.9 Free Mechanical Vibrations Spring-Mass Oscillator When the spring is not stretched and the mass m is at rest, the system is at equilibrium. Forces Acting in the System When the mass m is displaced
More informationExam II Review: Selected Solutions and Answers
November 9, 2011 Exam II Review: Selected Solutions and Answers NOTE: For additional worked problems see last year s review sheet and answers, the notes from class, and your text. Answers to problems from
More informationMath 4B Notes. Written by Victoria Kala SH 6432u Office Hours: T 12:45 1:45pm Last updated 7/24/2016
Math 4B Notes Written by Victoria Kala vtkala@math.ucsb.edu SH 6432u Office Hours: T 2:45 :45pm Last updated 7/24/206 Classification of Differential Equations The order of a differential equation is the
More informationSimple Harmonic Motion Practice Problems PSI AP Physics B
Simple Harmonic Motion Practice Problems PSI AP Physics B Name Multiple Choice 1. A block with a mass M is attached to a spring with a spring constant k. The block undergoes SHM. Where is the block located
More informationEXAM 2 MARCH 17, 2004
8.034 EXAM MARCH 7, 004 Name: Problem : /30 Problem : /0 Problem 3: /5 Problem 4: /5 Total: /00 Instructions: Please write your name at the top of every page of the exam. The exam is closed book, closed
More informationA: Brief Review of Ordinary Differential Equations
A: Brief Review of Ordinary Differential Equations Because of Principle # 1 mentioned in the Opening Remarks section, you should review your notes from your ordinary differential equations (odes) course
More informationYou may use a calculator, but you must show all your work in order to receive credit.
Math 2410-010/015 Exam II April 7 th, 2017 Name: Instructions: Key Answer each question to the best of your ability. All answers must be written clearly. Be sure to erase or cross out any work that you
More informationChapter 11 Vibrations and Waves
Chapter 11 Vibrations and Waves If an object vibrates or oscillates back and forth over the same path, each cycle taking the same amount of time, the motion is called periodic. The mass and spring system
More informationChapter 2: Complex numbers
Chapter 2: Complex numbers Complex numbers are commonplace in physics and engineering. In particular, complex numbers enable us to simplify equations and/or more easily find solutions to equations. We
More informationMATH 251 Examination I February 25, 2016 FORM A. Name: Student Number: Section:
MATH 251 Examination I February 25, 2016 FORM A Name: Student Number: Section: This exam has 13 questions for a total of 100 points. Show all your work! In order to obtain full credit for partial credit
More informationWave Phenomena Physics 15c
Wave Phenomena Physics 15c Lecture Harmonic Oscillators (H&L Sections 1.4 1.6, Chapter 3) Administravia! Problem Set #1! Due on Thursday next week! Lab schedule has been set! See Course Web " Laboratory
More information2nd-Order Linear Equations
4 2nd-Order Linear Equations 4.1 Linear Independence of Functions In linear algebra the notion of linear independence arises frequently in the context of vector spaces. If V is a vector space over the
More informationspring mass equilibrium position +v max
Lecture 20 Oscillations (Chapter 11) Review of Simple Harmonic Motion Parameters Graphical Representation of SHM Review of mass-spring pendulum periods Let s review Simple Harmonic Motion. Recall we used
More informationComputer Problems for Methods of Solving Ordinary Differential Equations
Computer Problems for Methods of Solving Ordinary Differential Equations 1. Have a computer make a phase portrait for the system dx/dt = x + y, dy/dt = 2y. Clearly indicate critical points and separatrices.
More informationWave Phenomena Physics 15c
Wave Phenomena Physics 15c Lecture Harmonic Oscillators (H&L Sections 1.4 1.6, Chapter 3) Administravia! Problem Set #1! Due on Thursday next week! Lab schedule has been set! See Course Web " Laboratory
More informationMAT187H1F Lec0101 Burbulla
Spring 2017 Second Order Linear Homogeneous Differential Equation DE: A(x) d 2 y dx 2 + B(x)dy dx + C(x)y = 0 This equation is called second order because it includes the second derivative of y; it is
More informationChapter 7. Homogeneous equations with constant coefficients
Chapter 7. Homogeneous equations with constant coefficients It has already been remarked that we can write down a formula for the general solution of any linear second differential equation y + a(t)y +
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 informationAdditional Homework Problems
Additional Homework Problems These problems supplement the ones assigned from the text. Use complete sentences whenever appropriate. Use mathematical terms appropriately. 1. What is the order of a differential
More information144 Chapter 3. Second Order Linear Equations
144 Chapter 3. Second Order Linear Equations PROBLEMS In each of Problems 1 through 8 find the general solution of the given differential equation. 1. y + 2y 3y = 0 2. y + 3y + 2y = 0 3. 6y y y = 0 4.
More informationMath 216 Second Midterm 16 November, 2017
Math 216 Second Midterm 16 November, 2017 This sample exam is provided to serve as one component of your studying for this exam in this course. Please note that it is not guaranteed to cover the material
More informationSection 9.8 Higher Order Linear Equations
Section 9.8 Higher Order Linear Equations Key Terms: Higher order linear equations Equivalent linear systems for higher order equations Companion matrix Characteristic polynomial and equation A linear
More informationLinear Independence. MATH 322, Linear Algebra I. J. Robert Buchanan. Spring Department of Mathematics
Linear Independence MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Given a set of vectors {v 1, v 2,..., v r } and another vector v span{v 1, v 2,...,
More informationMA 266 Review Topics - Exam # 2 (updated)
MA 66 Reiew Topics - Exam # updated Spring First Order Differential Equations Separable, st Order Linear, Homogeneous, Exact Second Order Linear Homogeneous with Equations Constant Coefficients The differential
More informationODE Math 3331 (Summer 2014) June 16, 2014
Page 1 of 12 Please go to the next page... Sample Midterm 1 ODE Math 3331 (Summer 2014) June 16, 2014 50 points 1. Find the solution of the following initial-value problem 1. Solution (S.O.V) dt = ty2,
More informationHigher Order Linear Equations Lecture 7
Higher Order Linear Equations Lecture 7 Dibyajyoti Deb 7.1. Outline of Lecture General Theory of nth Order Linear Equations. Homogeneous Equations with Constant Coefficients. 7.2. General Theory of nth
More informationNumerical Methods for the Solution of Differential Equations
Numerical Methods for the Solution of Differential Equations Markus Grasmair Vienna, winter term 2011 2012 Analytical Solutions of Ordinary Differential Equations 1. Find the general solution of the differential
More informationOrdinary Differential Equations
Ordinary Differential Equations for Engineers and Scientists Gregg Waterman Oregon Institute of Technology c 2017 Gregg Waterman This work is licensed under the Creative Commons Attribution 4.0 International
More informationMath 215/255 Final Exam (Dec 2005)
Exam (Dec 2005) Last Student #: First name: Signature: Circle your section #: Burggraf=0, Peterson=02, Khadra=03, Burghelea=04, Li=05 I have read and understood the instructions below: Please sign: Instructions:.
More informationHarmonic Motion: Exercises
Harmonic Motion: Exercises 1. The following is a list of forces, each of which is the net external force acting on an object with mass number m that is free to move in onedimension only. Assume that s
More information4 A mass-spring oscillating system undergoes SHM with a period T. What is the period of the system if the amplitude is doubled?
Slide 1 / 52 1 A block with a mass M is attached to a spring with a spring constant k. The block undergoes SHM. Where is the block located when its velocity is a maximum in magnitude? A 0 B + or - A C
More informationHigher-order differential equations
Higher-order differential equations Peyam Tabrizian Wednesday, November 16th, 2011 This handout is meant to give you a couple more example of all the techniques discussed in chapter 6, to counterbalance
More informationChapter 3: Second Order Equations
Exam 2 Review This review sheet contains this cover page (a checklist of topics from Chapters 3). Following by all the review material posted pertaining to chapter 3 (all combined into one file). Chapter
More informationHIGHER-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS II: Nonhomogeneous Case and Vibrations
HIGHER-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS II: Nonhomogeneous Case and Vibrations David Levermore Department of Mathematics University of Maryland 22 October 28 Because the presentation of this
More informationNotes 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 informationUnforced Mechanical Vibrations
Unforced Mechanical Vibrations Today we begin to consider applications of second order ordinary differential equations. 1. Spring-Mass Systems 2. Unforced Systems: Damped Motion 1 Spring-Mass Systems We
More informationJim Lambers MAT 285 Spring Semester Practice Exam 2 Solution. y(t) = 5 2 e t 1 2 e 3t.
. Solve the initial value problem which factors into Jim Lambers MAT 85 Spring Semester 06-7 Practice Exam Solution y + 4y + 3y = 0, y(0) =, y (0) =. λ + 4λ + 3 = 0, (λ + )(λ + 3) = 0. Therefore, the roots
More informationWork sheet / Things to know. Chapter 3
MATH 251 Work sheet / Things to know 1. Second order linear differential equation Standard form: Chapter 3 What makes it homogeneous? We will, for the most part, work with equations with constant coefficients
More informationMATH 135: COMPLEX NUMBERS
MATH 135: COMPLEX NUMBERS (WINTER, 010) The complex numbers C are important in just about every branch of mathematics. These notes 1 present some basic facts about them. 1. The Complex Plane A complex
More informationSolutions to Final Exam Sample Problems, Math 246, Fall 2013
Solutions to Final Exam Sample Problems, Math 246, Fall 23 () Consider the differential equation dy dt = (9 y2 )y 2 (a) Identify its stationary points and classify their stability (b) Sketch its phase-line
More informationWeek 9 solutions. k = mg/l = /5 = 3920 g/s 2. 20u + 400u u = 0,
Week 9 solutions ASSIGNMENT 20. (Assignment 19 had no hand-graded component.) 3.7.9. A mass of 20 g stretches a spring 5 cm. Suppose that the mass is also attached to a viscous damper with a damping constant
More informationDifferential Equations 2280 Sample Midterm Exam 3 with Solutions Exam Date: 24 April 2015 at 12:50pm
Differential Equations 228 Sample Midterm Exam 3 with Solutions Exam Date: 24 April 25 at 2:5pm Instructions: This in-class exam is 5 minutes. No calculators, notes, tables or books. No answer check is
More information2.3 Oscillation. The harmonic oscillator equation is the differential equation. d 2 y dt 2 r y (r > 0). Its solutions have the form
2. Oscillation So far, we have used differential equations to describe functions that grow or decay over time. The next most common behavior for a function is to oscillate, meaning that it increases and
More informationMath 304 Answers to Selected Problems
Math Answers to Selected Problems Section 6.. Find the general solution to each of the following systems. a y y + y y y + y e y y y y y + y f y y + y y y + 6y y y + y Answer: a This is a system of the
More informationMATH 251 Final Examination May 3, 2017 FORM A. Name: Student Number: Section:
MATH 5 Final Examination May 3, 07 FORM A Name: Student Number: Section: This exam has 6 questions for a total of 50 points. In order to obtain full credit for partial credit problems, all work must be
More informationTable of contents. d 2 y dx 2, As the equation is linear, these quantities can only be involved in the following manner:
M ath 0 1 E S 1 W inter 0 1 0 Last Updated: January, 01 0 Solving Second Order Linear ODEs Disclaimer: This lecture note tries to provide an alternative approach to the material in Sections 4. 4. 7 and
More informationMidterm 1 NAME: QUESTION 1 / 10 QUESTION 2 / 10 QUESTION 3 / 10 QUESTION 4 / 10 QUESTION 5 / 10 QUESTION 6 / 10 QUESTION 7 / 10 QUESTION 8 / 10
Midterm 1 NAME: RULES: You will be given the entire period (1PM-3:10PM) to complete the test. You can use one 3x5 notecard for formulas. There are no calculators nor those fancy cellular phones nor groupwork
More informationSection 3.7: Mechanical and Electrical Vibrations
Section 3.7: Mechanical and Electrical Vibrations Second order linear equations with constant coefficients serve as mathematical models for mechanical and electrical oscillations. For example, the motion
More informationLinear Second Order ODEs
Chapter 3 Linear Second Order ODEs In this chapter we study ODEs of the form (3.1) y + p(t)y + q(t)y = f(t), where p, q, and f are given functions. Since there are two derivatives, we might expect that
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 informationForced Mechanical Vibrations
Forced Mechanical Vibrations Today we use methods for solving nonhomogeneous second order linear differential equations to study the behavior of mechanical systems.. Forcing: Transient and Steady State
More informationExam Basics. midterm. 1 There will be 9 questions. 2 The first 3 are on pre-midterm material. 3 The next 1 is a mix of old and new material.
Exam Basics 1 There will be 9 questions. 2 The first 3 are on pre-midterm material. 3 The next 1 is a mix of old and new material. 4 The last 5 questions will be on new material since the midterm. 5 60
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 4 The Method of Variation of Parameters Problem 4.1 Solve y
More informationOld Math 330 Exams. David M. McClendon. Department of Mathematics Ferris State University
Old Math 330 Exams David M. McClendon Department of Mathematics Ferris State University Last updated to include exams from Fall 07 Contents Contents General information about these exams 3 Exams from Fall
More informationProblem Score Possible Points Total 150
Math 250 Fall 2010 Final Exam NAME: ID No: SECTION: This exam contains 17 problems on 13 pages (including this title page) for a total of 150 points. There are 10 multiple-choice problems and 7 partial
More informationSecond In-Class Exam Solutions Math 246, Professor David Levermore Thursday, 31 March 2011
Second In-Class Exam Solutions Math 246, Professor David Levermore Thursday, 31 March 211 (1) [6] Give the interval of definition for the solution of the initial-value problem d 4 y dt 4 + 7 1 t 2 dy dt
More informationSecond-Order Linear Differential Equations C 2
C8 APPENDIX C Additional Topics in Differential Equations APPENDIX C. Second-Order Homogeneous Linear Equations Second-Order Linear Differential Equations Higher-Order Linear Differential Equations Application
More informationMathematical Models. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Spring Department of Mathematics
Mathematical Models MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 Ordinary Differential Equations The topic of ordinary differential equations (ODEs)
More informationDo not write below here. Question Score Question Score Question Score
MATH-2240 Friday, May 4, 2012, FINAL EXAMINATION 8:00AM-12:00NOON Your Instructor: Your Name: 1. Do not open this exam until you are told to do so. 2. This exam has 30 problems and 18 pages including this
More informationAPPM 2360: Midterm exam 3 April 19, 2017
APPM 36: Midterm exam 3 April 19, 17 On the front of your Bluebook write: (1) your name, () your instructor s name, (3) your lecture section number and (4) a grading table. Text books, class notes, cell
More informationMathematical Models. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Fall Department of Mathematics
Mathematical Models MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Ordinary Differential Equations The topic of ordinary differential equations (ODEs) is
More information