Exam II Review: Selected Solutions and Answers
|
|
- Camilla Henry
- 6 years ago
- Views:
Transcription
1 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 the text are in the back of the book, so I don t work most of these out here. Be sure you do all the review sheet problems in preparation for the exam! 1. Define (in the context of 2nd order, linear differential equations) the following concepts and give examples where appropriate. (a) Homogeneous and non-homogeneous 2nd order linear ordinary differential equation: General second order non-homogeneous equation Homogeneous equation: set g(t) = 0. p(t)y + q(t)y + p(t)y = g(t). (b) Linear independence: Two functions f, g defined on an interval I are called linearly dependent if there exist constants c 1, c 2 not both zero so that c 1 f(t) + c 2 g(t) = 0 holds for all t in I. If this is not the case, then the functions are called linearly dependent. (c) Fundamental set of solutions: A fundamental set of solutions for a second order homogeneous equation is a pair of functions y 1 (t), y 2 (t) which are both solutions to the ODE and which and are linearly independent. By forming the linear combination y(t) = c 1 y 1 (t) + c 2 y 2 (t) from the solutions in the fundamental set, we can obtain the solution that satisfies any prescribed initial conditions by choosing the constants c 1, c 2. It does NOT make sense to talk about a fundamental set for a non-homogeneous equation. (d) The general solution to the homogeneous equation y(t) = c 1 y 1 (t) + c 2 y 2 (t) where y 1, y 2 are a fundamental set. The general solution contains all possible solutions to the ODE in the sense that given any initial condition, we can obtain the unique solution to the initial value problem by choosing the constants c 1, c 2 correctly. General solution to the non-homogeneous equation (see text 3.6): y(t) = c 1 y 1 (t) + c 2 y 2 (t) + Y (t). The combination c 1 y 1 (t) + c 2 y 2 (t) is the general solution to corresponding homogeneous equation and Y (t) is a particular solution to the non-homogeneous equation. Again, this is a general solution since we can obtain the solution to any initial value problem corresponding to this ODE by choosing the correct c 1, c 2. (e) Natural frequency and period. Quasi-period and frequency: See text chapter 3.8. (f) Resonance and amplitude modulation: See text chapter See chapters 3.1,3.4, and Give an example of two functions which are linearly independent on (, ) and two functions which are linearly dependent on (, ). Solution. Two linearly independent functions are f(t) = sin t g(t) = cos t and two linearly dependent functions are f(t) = 51! cos t + 94 π cos t. It turns out that sets of two functions are linearly dependent if and only if one is a constant multiple of the other (why?). For larger sets of functions you need the more careful definition given in 3.3.
2 4. What does the Wronskian tell you about linear dependence/independence for solutions of a 2nd order linear homogeneous differential equation? What does it tell about the linear dependence/independence of arbitrary functions? Use examples to explain your answer. Solution. For solutions to the ODE, the functions are linearly independent if and only if the Wronskian is not zero for some point t 0 in the interval. (See Theorem and remarks on 157). For arbitrary functions (not connected to any kind of differential equation), it is more complicated (see 3.3.1). In this case, the Wronskian of linearly dependent functions must be 0 for all t but it is NOT TRUE that if the Wronskian is 0 everywhere that the functions are linearly dependent. I leave the examples to you. 5. Are the functions f(t) = t and g(t) = t linearly independent on ( 1, 1)? On (0, 1)? On ( 1, 0)? Carefully explain. Solutions. Notice that if t < 0 then t = t whereas if t > 0 then t = t. The functions f(t) = t and g(t) = t are linearly dependent on (0, 1) since if we choose c 1 = 1 and c 2 = 1, we have that c 1 t + c 2 t = c 1 t + c 2 t = 0 holds for all t. Thus the functions are linearly dependent on this interval. This also true on ( 1, 0). Let c 1 = c 2 = 1: c 1 t + c 2 t = c 1 t c 2 t = 0 and this holds for all t on ( 1, 0). So the functions are linearly dependent on ( 1, 0) also. The functions must be linearly independent on ( 1, 1) however. One way to see this is to suppose that there are two constants c 1 and c 2 (not both zero) so that must hold for all t. c 1 t + c 2 t = 0 But if this is the case, then this holds for t = 1 and t = 1. In particular, we get that c 1 + c 2 = 0 c 1 + c 2 = 0 which immediately implies that c 1 = c 2 = 0. This is a contradiction so f, g must be linearly independent on this interval. 6. Find the general solution to each of the following homogeneous equations (a) 2y 13y + 6y = 0 Solution. Characteristic equation: Roots: r 1 = 6 and r 2 = 1/2. 2r 2 13r + 6 = 0 y(t) = c 1 e 6t + c 2 e t/2 (b) y 4y + 5y = 0 Solution. Characteristic equation: Roots: r 1 = 2 + i and r 2 = 2 i. r 2 4r + 5 = 0 y(t) = e 2t (c 1 cos t + c 2 sin t). (c) y 10y + 25y = 0
3 Solution. Characteristic equation: Roots: r = 5 (repeated). r 2 10r + 25 = 0 y(t) = c 1 e 5t + c 2 te 5t. 7. Describe how each of the solutions to the previous problems behave as t. Assume that y(0) = 0 and y (0) = 1 for each of the previous problems. Make a sketch of the solutions with these initial conditions. Proof. For the first problem, and y(0) = c 1 + c 2 = 0. Compute Solve the equations c 1 + c 2 = 0 and 6c 1 + c2 2 y(t) = c 1 e 6t + c 2 e t/2 y (t) = 6c 1 e 6t + c 2 2 et/2. y (0) = 6c 1 + c 2 2 = 1 = 1 to find that c 1 = 2/11 and c 2 = 2/11. Thus and y(t) as t. For the second problem, and y(0) = c 1 = 0. Compute Thus which oscillates and grows as t. For the third problem, and y(0) = c 1 = 0. Compute so that c 2 = 1/5. So and we note that y(t) as t. y(t) = 2 11 e6t 2 11 et/2 y(t) = e 2t (c 1 cos t + c 2 sin t) y (t) = 2c 2 e 2t sin t + c2e 2t cos t y (0) = c 2 = 1. y(t) = e 2t sin t y(t) = c 1 e 5t + c 2 te 5t y (t) = 5c 2 e 5t + c 2 te 5t y (0) = 5c 2 = 1 y(t) = t 5 e5t 8. Additional practice problems for homogeneous constant coefficient case: (a) 3.1: 9-11 (b) 3.4: 7-10 (c) 3.5: You must be able to use the reduction of order method. In this approach, you are given one solution y 1 (t) and must construct a second solution y 2 (t) by setting y 2 (t) = y 1 (t)v(t). You plug this back in to the original equation and find a first order equation for v. Some good practice problems are: 3.5:
4 3.6: #3 10. You must know how to use the method of undetermined coefficients: 3.6: 1-10 are good practice. I will certainly ask you to use this method for some easy non-homogeneous terms like g(t) = 3 cos 2t or g(t) = e t. 11. You must know how to use the variation of parameters formula. 3.7: 1-10 are good practice. 12. Spring-mass: , 3.9: 5,6,9. I will give you a spring mass problem (or part of one) to solve and interpret. It will be very similar to one of these problems. 13. Circuits: 3.8: 12, Some proofs; (a) Easy proofs: i. Suppose that y 1 and y 2 are a fundamental set for y + p(t)y + q(t)y = 0. Prove that c 1 y 1 (t) and c 2 y 2 (t) are a fundamental set. Proof. You must check that c 1 y 1, c 2 y 2 are both solutions to the differential equation and that they are linearly independent. Use the operator notation Now and by the linear properties of L, we have L[y] = y + p(t)y + q(t)y L[y 1 ] = L[y 2 ] = 0 L[c 1 y 1 ] = c 1 L[y 1 ] = 0. The same proof works for c 2 y 2. Therefore c 1 y 1, c 2 y 2 are both solutions to the differential equation. To verify that c 1 y 1, c 2 y 2 are linearly independent, you should compute the Wronskian. W (c 1 y 1, c 2 y 2 )(t) = c 1y 1 c 2 y 2 c 1 y 1 c 2 y 2 = c 1c 2 W (y 1, y 2 )(t) Now W (y 1, y 2 )(t) 0 since y 1, y 2 are linearly independent. Provided that c 1 and c 2 are not 0, then the solutions are linearly independent. Therefore, c 1 y 2, c 2 y 2 form a fundamental set. ii. Suppose p(t) and q(t) are continuous for all t. Find the unique solution to y + p(t)y + q(t)y = 0, y(0) = 0, y (0) = 0. Proof. We know that solutions to 2nd order linear ODE initial value problems with continuous coefficient functions are unique (theorem 3.2.1). It clear that y(t) = 0 satisfies the ODE and also the initial conditions. Therefore, it is the unique solution. iii. Prove that the sum of any two solutions to a 2nd order linear homogeneous equation is a solution to the same equation. Proof. General homogeneous linear 2nd order ODE: y + p(t)y + q(t)y = 0. Assume y 1, y 2 are solutions. Then prove this result by plugging the sum y = y 1 + y 2 into the differential equation and rearranging. iv. Suppose y 1 (t), y 2 (t) are solutions to a 2nd order linear homogeneous equation and that Y (t) is a solution to the corresponding non-homogeneous equation. Prove that is a solution the non-homogeneous equation. y = c 1 y 1 (t) + c 2 y 2 (t) + Y (t) Proof. General non-homogeneous linear 2nd order ODE: y + p(t)y + q(t)y = g(t). Prove this result by substituting y(t) into the differential equation and rearranging.
5 v. Suppose y 1 (t) is a solution to y + p(t)y + q(t)y = 0. Prove that y 2 (t) = cy 1 (t) is also solution for any constant c. Does y 1 (t), y 2 (t) constitute a fundamental set? Explain. Proof. Plug y 2 (t) into the equation to verify that it is a solution. y 1 (t) and y 2 (t) are NOT linearly independent however because one is a multiple of the other! So these are not a fundamental set. vi. The difference of two solutions to a 2nd order linear non-homogeneous equation is a solution to the corresponding homogeneous equation. Proof. Consider the non-homogeneous equation: If Y 1 (t) and Y 2 (t) are solutions to this equation y + p(t)y + q(t)y = g(t). Y 1 + p(t)y 1y + q(t)y = g(t) Y 2 + p(t)y 2 + q(t)y 2 = g(t). Subtract these two equations and simplify to obtain (Y 2 Y 1 ) + p(t)(y 2 Y 1 ) + q(t)(y 2 Y 1 ) = 0. Therefore the difference of two solutions to the non-homogeneous equation is a solution the corresponding homogeneous equation. (b) Harder proofs: i. Prove Abel s formula (Theorem 3.3.2): In text and notes. ii. Prove the variation of parameters formula: In text (Theorem 3.7.1) and notes. iii. Consider the equation ay + by + cy = 0. Suppose that the corresponding characteristic equation has one repeated root r. Prove y 1 (t) = e rt is a solution to the differential equation. Then use variation of parameters to construct a second solution: Done in class notes. iv. Use the series for e x to prove that Proof. Recall from calculus II that e (λ+µi)t = e λt (cos µt + i sin µt) e x = and that this series converges for all values of x. We will assume two facts which require some more advanced mathematics to carefully establish: A. That e (λ+iµ)t = e λt e iµt B. That we can replace x in the power series with iµt and still have a convergent series. We have e (λ+µi)t = e λt e iµt ( ) = e λt (iµt) n n! ( ) = e λt ( 1) n (µt) 2n ( 1) n 1 (µt) 2n 1 + i (2n)! (2n 1)! = e λt (cos µt + i sin µt) It might have been helpful in doing this proof to look up the cosine and sine power series. Also, its a good idea to write out the first few terms of each of the series appearing above if your not quite sure how to get the general pattern for the terms. x n n! n=1
Chapter 4: Higher Order Linear Equations
Chapter 4: Higher Order Linear Equations MATH 351 California State University, Northridge April 7, 2014 MATH 351 (Differential Equations) Ch 4 April 7, 2014 1 / 11 Sec. 4.1: General Theory of nth Order
More informationWorksheet # 2: Higher Order Linear ODEs (SOLUTIONS)
Name: November 8, 011 Worksheet # : Higher Order Linear ODEs (SOLUTIONS) 1. A set of n-functions f 1, f,..., f n are linearly independent on an interval I if the only way that c 1 f 1 (t) + c f (t) +...
More informationSecond Order Linear Equations
October 13, 2016 1 Second And Higher Order Linear Equations In first part of this chapter, we consider second order linear ordinary linear equations, i.e., a differential equation of the form L[y] = d
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 informationLinear Independence and the Wronskian
Linear Independence and the Wronskian MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 Operator Notation Let functions p(t) and q(t) be continuous functions
More informationMATH 308 Differential Equations
MATH 308 Differential Equations Summer, 2014, SET 5 JoungDong Kim Set 5: Section 3.1, 3.2 Chapter 3. Second Order Linear Equations. Section 3.1 Homogeneous Equations with Constant Coefficients. In this
More informationMATH 308 Differential Equations
MATH 308 Differential Equations Summer, 2014, SET 6 JoungDong Kim Set 6: Section 3.3, 3.4, 3.5, 3.6 Section 3.3 Complex Roots of the Characteristic Equation Recall that a second order ODE with constant
More informationLinear Homogeneous ODEs of the Second Order with Constant Coefficients. Reduction of Order
Linear Homogeneous ODEs of the Second Order with Constant Coefficients. Reduction of Order October 2 6, 2017 Second Order ODEs (cont.) Consider where a, b, and c are real numbers ay +by +cy = 0, (1) Let
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 information1 Continuation of solutions
Math 175 Honors ODE I Spring 13 Notes 4 1 Continuation of solutions Theorem 1 in the previous notes states that a solution to y = f (t; y) (1) y () = () exists on a closed interval [ h; h] ; under certain
More informationDifferential Equations Practice: 2nd Order Linear: Nonhomogeneous Equations: Undetermined Coefficients Page 1
Differential Equations Practice: 2nd Order Linear: Nonhomogeneous Equations: Undetermined Coefficients Page 1 Questions Example (3.5.3) Find a general solution of the differential equation y 2y 3y = 3te
More informationSecond Order Differential Equations Lecture 6
Second Order Differential Equations Lecture 6 Dibyajyoti Deb 6.1. Outline of Lecture Repeated Roots; Reduction of Order Nonhomogeneous Equations; Method of Undetermined Coefficients Variation of Parameters
More informationµ = e R p(t)dt where C is an arbitrary constant. In the presence of an initial value condition
MATH 3860 REVIEW FOR FINAL EXAM The final exam will be comprehensive. It will cover materials from the following sections: 1.1-1.3; 2.1-2.2;2.4-2.6;3.1-3.7; 4.1-4.3;6.1-6.6; 7.1; 7.4-7.6; 7.8. The following
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 informationNonhomogeneous Equations and Variation of Parameters
Nonhomogeneous Equations Variation of Parameters June 17, 2016 1 Nonhomogeneous Equations 1.1 Review of First Order Equations If we look at a first order homogeneous constant coefficient ordinary differential
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 informationSecond-Order Linear ODEs
Second-Order Linear ODEs A second order ODE is called linear if it can be written as y + p(t)y + q(t)y = r(t). (0.1) It is called homogeneous if r(t) = 0, and nonhomogeneous otherwise. We shall assume
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 informationSecond order linear equations
Second order linear equations Samy Tindel Purdue University Differential equations - MA 266 Taken from Elementary differential equations by Boyce and DiPrima Samy T. Second order equations Differential
More informationProblem Score Possible Points Total 150
Math 250 Spring 2010 Final Exam NAME: ID No: SECTION: This exam contains 17 problems on 14 pages (including this title page) for a total of 150 points. The exam has a multiple choice part, and partial
More informationNon-homogeneous equations (Sect. 3.6).
Non-homogeneous equations (Sect. 3.6). We study: y + p(t) y + q(t) y = f (t). Method of variation of parameters. Using the method in an example. The proof of the variation of parameter method. Using the
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 informationHomogeneous Equations with Constant Coefficients
Homogeneous Equations with Constant Coefficients MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 General Second Order ODE Second order ODEs have the form
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 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 informationMath 256: Applied Differential Equations: Final Review
Math 256: Applied Differential Equations: Final Review Chapter 1: Introduction, Sec 1.1, 1.2, 1.3 (a) Differential Equation, Mathematical Model (b) Direction (Slope) Field, Equilibrium Solution (c) Rate
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 information6. Linear Differential Equations of the Second Order
September 26, 2012 6-1 6. Linear Differential Equations of the Second Order A differential equation of the form L(y) = g is called linear if L is a linear operator and g = g(t) is continuous. The most
More informationHW2 Solutions. MATH 20D Fall 2013 Prof: Sun Hui TA: Zezhou Zhang (David) October 14, Checklist: Section 2.6: 1, 3, 6, 8, 10, 15, [20, 22]
HW2 Solutions MATH 20D Fall 2013 Prof: Sun Hui TA: Zezhou Zhang (David) October 14, 2013 Checklist: Section 2.6: 1, 3, 6, 8, 10, 15, [20, 22] Section 3.1: 1, 2, 3, 9, 16, 18, 20, 23 Section 3.2: 1, 2,
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 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 20D: Form B Final Exam Dec.11 (3:00pm-5:50pm), Show all of your work. No credit will be given for unsupported answers.
Turn off and put away your cell phone. No electronic devices during the exam. No books or other assistance during the exam. Show all of your work. No credit will be given for unsupported answers. Write
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 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 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 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 information1 Some general theory for 2nd order linear nonhomogeneous
Math 175 Honors ODE I Spring, 013 Notes 5 1 Some general theory for nd order linear nonhomogeneous equations 1.1 General form of the solution Suppose that p; q; and g are continuous on an interval I; and
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 informationSMA 208: Ordinary differential equations I
SMA 208: Ordinary differential equations I First Order differential equations Lecturer: Dr. Philip Ngare (Contacts: pngare@uonbi.ac.ke, Tue 12-2 PM) School of Mathematics, University of Nairobi Feb 26,
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 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 informationLecture Notes for Math 251: ODE and PDE. Lecture 13: 3.4 Repeated Roots and Reduction Of Order
Lecture Notes for Math 251: ODE and PDE. Lecture 13: 3.4 Repeated Roots and Reduction Of Order Shawn D. Ryan Spring 2012 1 Repeated Roots of the Characteristic Equation and Reduction of Order Last Time:
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 informationUnderstand the existence and uniqueness theorems and what they tell you about solutions to initial value problems.
Review Outline To review for the final, look over the following outline and look at problems from the book and on the old exam s and exam reviews to find problems about each of the following topics.. Basics
More informationMATH 251 Examination I July 1, 2013 FORM A. Name: Student Number: Section:
MATH 251 Examination I July 1, 2013 FORM A Name: Student Number: Section: This exam has 12 questions for a total of 100 points. Show all your work! In order to obtain full credit for partial credit problems,
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 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 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 informationPartial proof: y = ϕ 1 (t) is a solution to y + p(t)y = 0 implies. Thus y = cϕ 1 (t) is a solution to y + p(t)y = 0 since
Existence and Uniqueness for LINEAR DEs. Homogeneous: y (n) + p 1 (t)y (n 1) +...p n 1 (t)y + p n (t)y = 0 Non-homogeneous: g(t) 0 y (n) + p 1 (t)y (n 1) +...p n 1 (t)y + p n (t)y = g(t) 1st order LINEAR
More informationFirst and Second Order Differential Equations Lecture 4
First and Second Order Differential Equations Lecture 4 Dibyajyoti Deb 4.1. Outline of Lecture The Existence and the Uniqueness Theorem Homogeneous Equations with Constant Coefficients 4.2. The Existence
More informationMath 216 Second Midterm 19 March, 2018
Math 26 Second Midterm 9 March, 28 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 that
More information1. Why don t we have to worry about absolute values in the general form for first order differential equations with constant coefficients?
1. Why don t we have to worry about absolute values in the general form for first order differential equations with constant coefficients? Let y = ay b with y(0) = y 0 We can solve this as follows y =
More informationSecond Order Linear Equations
Second Order Linear Equations Linear Equations The most general linear ordinary differential equation of order two has the form, a t y t b t y t c t y t f t. 1 We call this a linear equation because the
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 informationStudy guide - Math 220
Study guide - Math 220 November 28, 2012 1 Exam I 1.1 Linear Equations An equation is linear, if in the form y + p(t)y = q(t). Introducing the integrating factor µ(t) = e p(t)dt the solutions is then in
More informationProblem 1 (Equations with the dependent variable missing) By means of the substitutions. v = dy dt, dv
V Problem 1 (Equations with the dependent variable missing) By means of the substitutions v = dy dt, dv dt = d2 y dt 2 solve the following second-order differential equations 1. t 2 d2 y dt + 2tdy 1 =
More information2. Higher-order Linear ODE s
2. Higher-order Linear ODE s 2A. Second-order Linear ODE s: General Properties 2A-1. On the right below is an abbreviated form of the ODE on the left: (*) y + p()y + q()y = r() Ly = r() ; where L is the
More informationEntrance Exam, Differential Equations April, (Solve exactly 6 out of the 8 problems) y + 2y + y cos(x 2 y) = 0, y(0) = 2, y (0) = 4.
Entrance Exam, Differential Equations April, 7 (Solve exactly 6 out of the 8 problems). Consider the following initial value problem: { y + y + y cos(x y) =, y() = y. Find all the values y such that the
More informationHonors Differential Equations
MIT OpenCourseWare http://ocw.mit.edu 18.034 Honors Differential Equations Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. LECTURE 13. INHOMOGENEOUS
More informationMath 23 Practice Quiz 2018 Spring
1. Write a few examples of (a) a homogeneous linear differential equation (b) a non-homogeneous linear differential equation (c) a linear and a non-linear differential equation. 2. Calculate f (t). Your
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 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 informationSign the pledge. On my honor, I have neither given nor received unauthorized aid on this Exam : 11. a b c d e. 1. a b c d e. 2.
Math 258 Name: Final Exam Instructor: May 7, 2 Section: Calculators are NOT allowed. Do not remove this answer page you will return the whole exam. You will be allowed 2 hours to do the test. You may leave
More informationHigher Order Linear Equations
C H A P T E R 4 Higher Order Linear Equations 4.1 1. The differential equation is in standard form. Its coefficients, as well as the function g(t) = t, are continuous everywhere. Hence solutions are valid
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 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 informationLINEAR EQUATIONS OF HIGHER ORDER. EXAMPLES. General framework
Differential Equations Grinshpan LINEAR EQUATIONS OF HIGHER ORDER. EXAMPLES. We consider linear ODE of order n: General framework (1) x (n) (t) + P n 1 (t)x (n 1) (t) + + P 1 (t)x (t) + P 0 (t)x(t) = 0
More informationSection 6.4 DEs with Discontinuous Forcing Functions
Section 6.4 DEs with Discontinuous Forcing Functions Key terms/ideas: Discontinuous forcing function in nd order linear IVPs Application of Laplace transforms Comparison to viewing the problem s solution
More information20D - Homework Assignment 5
Brian Bowers TA for Hui Sun MATH D Homework Assignment 5 November 8, 3 D - Homework Assignment 5 First, I present the list of all matrix row operations. We use combinations of these steps to row reduce
More informationFind the Fourier series of the odd-periodic extension of the function f (x) = 1 for x ( 1, 0). Solution: The Fourier series is.
Review for Final Exam. Monday /09, :45-:45pm in CC-403. Exam is cumulative, -4 problems. 5 grading attempts per problem. Problems similar to homeworks. Integration and LT tables provided. No notes, no
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 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 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 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 informationSection 3.1 Second Order Linear Homogeneous DEs with Constant Coefficients
Section 3. Second Order Linear Homogeneous DEs with Constant Coefficients Key Terms/ Ideas: Initial Value Problems Homogeneous DEs with Constant Coefficients Characteristic equation Linear DEs of second
More informationSeries Solutions Near an Ordinary Point
Series Solutions Near an Ordinary Point MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 Ordinary Points (1 of 2) Consider the second order linear homogeneous
More informationLecture 2. Classification of Differential Equations and Method of Integrating Factors
Math 245 - Mathematics of Physics and Engineering I Lecture 2. Classification of Differential Equations and Method of Integrating Factors January 11, 2012 Konstantin Zuev (USC) Math 245, Lecture 2 January
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 informationLecture 9. Scott Pauls 1 4/16/07. Dartmouth College. Math 23, Spring Scott Pauls. Last class. Today s material. Group work.
Lecture 9 1 1 Department of Mathematics Dartmouth College 4/16/07 Outline Repeated Roots Repeated Roots Repeated Roots Material from last class Wronskian: linear independence Constant coeffecient equations:
More informationExistence Theory: Green s Functions
Chapter 5 Existence Theory: Green s Functions In this chapter we describe a method for constructing a Green s Function The method outlined is formal (not rigorous) When we find a solution to a PDE by constructing
More informationLinear algebra and differential equations (Math 54): Lecture 19
Linear algebra and differential equations (Math 54): Lecture 19 Vivek Shende April 5, 2016 Hello and welcome to class! Previously We have discussed linear algebra. This time We start studying differential
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 informationHomework 9 - Solutions. Math 2177, Lecturer: Alena Erchenko
Homework 9 - Solutions Math 2177, Lecturer: Alena Erchenko 1. Classify the following differential equations (order, determine if it is linear or nonlinear, if it is linear, then determine if it is homogeneous
More informationReview of Lecture 9 Existence and Uniqueness
Review of Lecture 9 Existence and Uniqueness We consider y = f (x, y) with a given initial condition y(x 0 ) = y 0. There is a solution passing through (x 0, y 0 ). It is defined on some interval (a, b)
More informationLecture 17: Nonhomogeneous equations. 1 Undetermined coefficients method to find a particular
Lecture 17: Nonhomogeneous equations 1 Undetermined coefficients method to find a particular solution The method of undetermined coefficients (sometimes referred to as the method of justicious guessing)
More informationSolutions to Math 53 Math 53 Practice Final
Solutions to Math 5 Math 5 Practice Final 20 points Consider the initial value problem y t 4yt = te t with y 0 = and y0 = 0 a 8 points Find the Laplace transform of the solution of this IVP b 8 points
More informationMath 23: Differential Equations (Winter 2017) Midterm Exam Solutions
Math 3: Differential Equations (Winter 017) Midterm Exam Solutions 1. [0 points] or FALSE? You do not need to justify your answer. (a) [3 points] Critical points or equilibrium points for a first order
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 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 informationThe Laplace Transform
C H A P T E R 6 The Laplace Transform Many practical engineering problems involve mechanical or electrical systems acted on by discontinuous or impulsive forcing terms. For such problems the methods described
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 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 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 informationMath 216 Second Midterm 28 March, 2013
Math 26 Second Midterm 28 March, 23 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 that
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 informationAMATH 351 Mar 15, 2013 FINAL REVIEW. Instructor: Jiri Najemnik
AMATH 351 Mar 15, 013 FINAL REVIEW Instructor: Jiri Najemni ABOUT GRADES Scores I have so far will be posted on the website today sorted by the student number HW4 & Exam will be added early next wee Let
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 informationSection 4.7: Variable-Coefficient Equations
Cauchy-Euler Equations Section 4.7: Variable-Coefficient Equations Before concluding our study of second-order linear DE s, let us summarize what we ve done. In Sections 4.2 and 4.3 we showed how to find
More informationOrdinary Differential Equations
II 12/01/2015 II Second order linear equations with constant coefficients are important in two physical processes, namely, Mechanical and Electrical oscillations. Actually from the Math point of view,
More informationChapter 2: First Order DE 2.4 Linear vs. Nonlinear DEs
Chapter 2: First Order DE 2.4 Linear vs. Nonlinear DEs First Order DE 2.4 Linear vs. Nonlinear DE We recall the general form of the First Oreder DEs (FODE): dy = f(t, y) (1) dt where f(t, y) is a function
More information