Name: Solutions Exam 3
|
|
- Austen James
- 5 years ago
- Views:
Transcription
1 Instructions. Answer each of the questions on your own paper. Put your name on each page of your paper. Be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without supporting work. A table of Laplace transforms has been appended to the exam. The following trigonometric identities may also be of use: sin(θ+ϕ) = sinθcosϕ+sinϕcosθ cos(θ+ϕ) = cosθcosϕ sinθsinϕ. [20 Points] Find the general solution of the following Cauchy-Euler equations: (a) 3t 2 y 7ty +3y = 0. Solution. The indicial polynomial is Q(s) = 3s(s ) 7s+3 = 3s 2 0s+3 = (3s )(s 3), which has the two distinct real roots /3 and 3. Hence the general solution is (b) t 2 y ty +0y = 0. Solution. The indicial polynomial is y = c t /3 +c 2 t 3. Q(s) = s(s ) s+ = s 2 2s+0 = (s ) 2 +9, which has the complex roots ±3i. Hence the general solution is y = c tcos(3ln t )+c 2 tsin(3ln t ). 2. [20 Points] Use variation of parameters to find a particular solution of the nonhomogeneous differential equation y 4y +4y = t /2 e 2t. You may assume that the solution of the homogeneous equation y 4y +4y = 0 is y h = c e 2t +c 2 te 2t. Solution. Letting y = e 2t and y 2 = te 2t, a particular solution has the form y p = u y +u 2 y 2 = u e 2t +u 2 te 2t, Math 2065 Section 2 November 2, 208
2 where u and u 2 are unknown functions whose derivatives satisfy the simultaneous equations u e2t +u 2 te2t = 0 u (2e 2t +u 2(e 2t +2te 2t ) = t /2 e 2t. Dividing both equations by e 2t gives the simpler equations u +u 2 t = 0 2u +u 2 (+2t) = t/2. Applying Cramer s rule gives 0 t u = t /2 +2t t 2 +2t and 0 u 2 t /2 2 = t 2 +2t = t3/2 = t 3/2 = t/2 = t/2 Integrating gives u = 2 5 t5/2 and u 2 = 2 3 t3/2 so that y p = 2 5 t5/2 e 2t t3/2 te 2t = 4 5 t5/2 e 2t. 3. [20 Points] Let f be the function defined by t 2 if 0 t < 2, f(t) = 3 if t 2. (a) Sketch the graph of f(t) over the interval [0, 4] Math 2065 Section 2 November 2, 208 2
3 (b) Find the Laplace transform of f(t). Solution. Use characteristic functions to write f(t) in terms of unit step functions: f(t) = t 2 χ [0,2) (t)+3χ [2, ) (t) = t 2 (h(t) h(t 2))+3h(t 2) = t 2 +(3 t 2 )h(t 2). Now apply the second translation theorem to get F(s) = Lf(t)} = 2 s 3 +e 2s L 3 (t+2) 2} = 2 s 3 +e 2s L 3 (t 2 +4t+4) } = 2 s 3 +e 2s L t 2 4t ) } = 2 s 3 e 2s ( 2 s s 2 + s 4. [20 Points] Find the inverse Laplace transform of the following functions: (a) F(s) = e 4s (s+2) 3 Solution. F(s) = F (s)e 4s where F (s) = ). (s+2) 3. Let f (t) = L F (s)} = 2 t2 e 2t. Then the inverse of the second translation theorem gives f(t) = L F(s)} = f (t 4)h(t 4) (b) G(s) = 2s s 2 +2s+5 e 2s = 2 (t 4)2 e 2(t 4) h(t 4) 0 if 0 t < 4 = 2 (t 4)2 e 2(t 4) if t 4. Math 2065 Section 2 November 2, 208 3
4 Solution. Let G (s) = G (s) = 2s s 2 +2s+5. Then 2s s 2 +2s+5 = 2s (s+) 2 +4 = 2(s+) 2 (s+) 2 +4 = 2(s+) (s+) (s+) Thus, taking the inverse Laplace transform gives g (t) = L G (s)} = 2e t cos2t e t sin2t, so that the inverse of the second translation theorem gives g(t) = L G(s)} = g (t 2)h(t 2) ( ) = 2e (t 2) cos2(t 2) e (t 2) sin2(t 2) h(t 2) 0 if 0 t < 2 = 2e (t 2) cos2(t 2) e (t 2) sin2(t 2) if t [20 Points] Solve the following initial value problem: y +y = δ(t π), y(0) = 0, y (0) =. (Remember that δ(t c) is the Dirac delta function centered at c.) Give a careful sketch of the graph of the solution for the interval 0 t 2π. Solution. Let Y(s) = Ly(t)} be the Laplace transform of the solution function. Apply the Laplace transform to both sides of the equation to get Thus, s 2 Y(s) +Y(s) = e πs. (s 2 +)Y(s) = +e πs, and hence Y(s) = s 2 + +e πs s 2 +. Apply the inverse Laplace transform to get y(t) = sint+h(t π)sin(t π) = sint (sint)h(t π) sint if 0 t < π, = 0 if t π. Math 2065 Section 2 November 2, 208 4
5 π 2π 3π Graph of y(t). Math 2065 Section 2 November 2, 208 5
6 Laplace Transform Table f(t) F(s) = Lf(t)}(s). 2. t n 3. e at 4. t n e at 5. cos bt 6. sin bt 7. e at cosbt 8. e at sinbt 9. h(t c) s n! s n+ s a n! (s a) n+ s s 2 +b 2 b s 2 +b 2 s a (s a) 2 +b 2 b (s a) 2 +b 2 e sc 0. δ c (t) = δ(t c) e sc s Laplace Transform Principles Linearity Input Derivative Principles First Translation Principle Transform Derivative Principle Laf(t)+bg(t)} = alf}+blg} Lf (t)}(s) = slf(t)} f(0) Lf (t)}(s) = s 2 Lf(t)} sf(0) f (0) Le at f(t)} = F(s a) L tf(t)}(s) = d ds F(s) Second Translation Principle Lh(t c)f(t c)} = e sc F(s), or Lg(t)h(t c)} = e sc Lg(t+c)}. Math 2065 Section 2 November 2, 208 6
7 Partial Fraction Expansion Theorems The following two theorems are the main partial fractions expansion theorems, as presented in the text. Theorem (Linear Case). Suppose a proper rational function can be written in the form (s λ) n q(s) and q(λ) 0. Then there is a unique number A and a unique polynomial p (s) such that (s λ) n q(s) = A (s λ) + p (s) n (s λ) n q(s). () The number A and the polynomial p (s) are given by A = p 0(λ) q(λ) and p (s) = p 0(s) A q(s). (2) s λ Theorem 2 (Irreducible Quadratic Case). Suppose a real proper rational function can be written in the form (s 2 +cs+d) n q(s), where s 2 + cs + d is an irreducible quadratic that is factored completely out of q(s). Then there is a unique linear term B s+c and a unique polynomial p (s) such that (s 2 +cs+d) n q(s) = B s+c (s 2 +cs+d) n + p (s) (s s +cs+d) n q(s). (3) If a+ib is a complex root of s 2 +cs+d then B s+c and the polynomial p (s) are given by B (a+ib)+c = p 0(a+ib) q(a+ib) and p (s) = p 0(s) (B s+c )q(s). (4) s 2 +cs+d Math 2065 Section 2 November 2, 208 7
Name: Solutions Final Exam
Instructions. Answer each of the questions on your own paper. Put your name on each page of your paper. Be sure to show your work so that partial credit can be adequately assessed. Credit will not be given
More informationName: Solutions Exam 4
Instructions. Answer each of the questions on your own paper. Put your name on each page of your paper. Be sure to show your work so that partial credit can be adequately assessed. Credit will not be given
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 informationName: Solutions Exam 3
Intruction. Anwer each of the quetion on your own paper. Put your name on each page of your paper. Be ure to how your work o that partial credit can be adequately aeed. Credit will not be given for anwer
More informationMATH 251 Examination II April 4, 2016 FORM A. Name: Student Number: Section:
MATH 251 Examination II April 4, 2016 FORM A Name: Student Number: Section: This exam has 12 questions for a total of 100 points. In order to obtain full credit for partial credit problems, all work must
More informationName: Solutions Exam 2
Intruction. Anwer each of the quetion on your own paper. Put your name on each page of your paper. Be ure to how your work o that partial credit can be adequately aeed. Credit will not be given for anwer
More informationName: Solutions Exam 2
Name: Solution Exam Intruction. Anwer each of the quetion on your own paper. Put your name on each page of your paper. Be ure to how your work o that partial credit can be adequately aeed. Credit will
More informationMATH 251 Examination II April 3, 2017 FORM A. Name: Student Number: Section:
MATH 251 Examination II April 3, 2017 FORM A Name: Student Number: Section: This exam has 12 questions for a total of 100 points. In order to obtain full credit for partial credit problems, all work must
More informationExam 2 Study Guide: MATH 2080: Summer I 2016
Exam Study Guide: MATH 080: Summer I 016 Dr. Peterson June 7 016 First Order Problems Solve the following IVP s by inspection (i.e. guessing). Sketch a careful graph of each solution. (a) u u; u(0) 0.
More information8. [12 Points] Find a particular solution of the differential equation. t 2 y + ty 4y = t 3, y h = c 1 t 2 + c 2 t 2.
Intruction. Anwer each of the quetion on your own paper. Put your name on each page of your paper. Be ure to how your work o that partial credit can be adequately aeed. Credit will not be given for anwer
More informationMATH 251 Examination II April 7, 2014 FORM A. Name: Student Number: Section:
MATH 251 Examination II April 7, 2014 FORM A Name: Student Number: Section: This exam has 12 questions for a total of 100 points. In order to obtain full credit for partial credit problems, all work must
More informationSection 7.4: Inverse Laplace Transform
Section 74: Inverse Laplace Transform A natural question to ask about any function is whether it has an inverse function We now ask this question about the Laplace transform: given a function F (s), will
More information(an improper integral)
Chapter 7 Laplace Transforms 7.1 Introduction: A Mixing Problem 7.2 Definition of the Laplace Transform Def 7.1. Let f(t) be a function on [, ). The Laplace transform of f is the function F (s) defined
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 informationThe Laplace Transform and the IVP (Sect. 6.2).
The Laplace Transform and the IVP (Sect..2). Solving differential equations using L ]. Homogeneous IVP. First, second, higher order equations. Non-homogeneous IVP. Recall: Partial fraction decompositions.
More informationLaplace Transform. Chapter 4
Chapter 4 Laplace Transform It s time to stop guessing solutions and find a systematic way of finding solutions to non homogeneous linear ODEs. We define the Laplace transform of a function f in the following
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 informatione st f (t) dt = e st tf(t) dt = L {t f(t)} s
Additional operational properties How to find the Laplace transform of a function f (t) that is multiplied by a monomial t n, the transform of a special type of integral, and the transform of a periodic
More informationMATH 251 Examination II November 5, 2018 FORM A. Name: Student Number: Section:
MATH 251 Examination II November 5, 2018 FORM A Name: Student Number: Section: This exam has 14 questions for a total of 100 points. In order to obtain full credit for partial credit problems, all work
More informationMA26600 FINAL EXAM INSTRUCTIONS December 13, You must use a #2 pencil on the mark sense sheet (answer sheet).
MA266 FINAL EXAM INSTRUCTIONS December 3, 2 NAME INSTRUCTOR. You must use a #2 pencil on the mark sense sheet (answer sheet). 2. On the mark-sense sheet, fill in the instructor s name (if you do not know,
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 251 Final Examination December 16, 2015 FORM A. Name: Student Number: Section:
MATH 5 Final Examination December 6, 5 FORM A Name: Student Number: Section: This exam has 7 questions for a total of 5 points. In order to obtain full credit for partial credit problems, all work must
More informationLaplace Transforms Chapter 3
Laplace Transforms Important analytical method for solving linear ordinary differential equations. - Application to nonlinear ODEs? Must linearize first. Laplace transforms play a key role in important
More informationPractice Problems For Test 3
Practice Problems For Test 3 Power Series Preliminary Material. Find the interval of convergence of the following. Be sure to determine the convergence at the endpoints. (a) ( ) k (x ) k (x 3) k= k (b)
More information+ + LAPLACE TRANSFORM. Differentiation & Integration of Transforms; Convolution; Partial Fraction Formulas; Systems of DEs; Periodic Functions.
COLOR LAYER red LAPLACE TRANSFORM Differentiation & Integration of Transforms; Convolution; Partial Fraction Formulas; Systems of DEs; Periodic Functions. + Differentiation of Transforms. F (s) e st f(t)
More informationChapter DEs with Discontinuous Force Functions
Chapter 6 6.4 DEs with Discontinuous Force Functions Discontinuous Force Functions Using Laplace Transform, as in 6.2, we solve nonhomogeneous linear second order DEs with constant coefficients. The only
More informationMATH 251 Final Examination May 4, 2015 FORM A. Name: Student Number: Section:
MATH 251 Final Examination May 4, 2015 FORM A Name: Student Number: Section: This exam has 16 questions for a total of 150 points. In order to obtain full credit for partial credit problems, all work must
More informationMath 216 Second Midterm 20 March, 2017
Math 216 Second Midterm 20 March, 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 informationPractice Problems For Test 3
Practice Problems For Test 3 Power Series Preliminary Material. Find the interval of convergence of the following. Be sure to determine the convergence at the endpoints. (a) ( ) k (x ) k (x 3) k= k (b)
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 informationwe get y 2 5y = x + e x + C: From the initial condition y(0) = 1, we get 1 5 = 0+1+C; so that C = 5. Completing the square to solve y 2 5y = x + e x 5
Math 24 Final Exam Solution 17 December 1999 1. Find the general solution to the differential equation ty 0 +2y = sin t. Solution: Rewriting the equation in the form (for t 6= 0),we find that y 0 + 2 t
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 information37. f(t) sin 2t cos 2t 38. f(t) cos 2 t. 39. f(t) sin(4t 5) 40.
28 CHAPTER 7 THE LAPLACE TRANSFORM EXERCISES 7 In Problems 8 use Definition 7 to find {f(t)} 2 3 4 5 6 7 8 9 f (t),, f (t) 4,, f (t) t,, f (t) 2t,, f (t) sin t,, f (t), cos t, t t t 2 t 2 t t t t t t t
More informationHIGHER-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS IV: Laplace Transform Method David Levermore Department of Mathematics University of Maryland
HIGHER-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS IV: Laplace Transform Method David Levermore Department of Mathematics University of Maryland 9 December Because the presentation of this material in
More informationHIGHER-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS IV: Laplace Transform Method. David Levermore Department of Mathematics University of Maryland
HIGHER-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS IV: Laplace Transform Method David Levermore Department of Mathematics University of Maryland 6 April Because the presentation of this material in lecture
More informationComputing inverse Laplace Transforms.
Review Exam 3. Sections 4.-4.5 in Lecture Notes. 60 minutes. 7 problems. 70 grade attempts. (0 attempts per problem. No partial grading. (Exceptions allowed, ask you TA. Integration table included. Complete
More informationMath 353 Lecture Notes Week 6 Laplace Transform: Fundamentals
Math 353 Lecture Notes Week 6 Laplace Transform: Fundamentals J. Wong (Fall 217) October 7, 217 What did we cover this week? Introduction to the Laplace transform Basic theory Domain and range of L Key
More informationI have read and understood the instructions regarding academic dishonesty:
Name Final Exam MATH 6600 SPRING 08 MARK TEST 0 ON YOUR SCANTRON! Student ID Section Number (see list below 03 UNIV 03 0:30am TR Alper, Onur 04 REC 3:30pm MWF Luo, Tao 05 UNIV 03 :30pm TR Hora, Raphael
More informationHonors Differential Equations
MIT OpenCourseWare http://ocw.mit.edu 8.034 Honors Differential Equations Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. LECTURE 20. TRANSFORM
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 informationMath 216 Final Exam 14 December, 2017
Math 216 Final Exam 14 December, 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 that
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 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 informationMath 307 A - Spring 2015 Final Exam June 10, 2015
Name: Math 307 A - Spring 2015 Final Exam June 10, 2015 Student ID Number: There are 8 pages of questions. In addition, the last page is the basic Laplace transform table. Make sure your exam contains
More informationName: ID.NO: Fall 97. PLEASE, BE NEAT AND SHOW ALL YOUR WORK; CIRCLE YOUR ANSWER. NO NOTES, BOOKS, CALCULATORS, TAPE PLAYERS, or COMPUTERS.
MATH 303-2/6/97 FINAL EXAM - Alternate WILKERSON SECTION Fall 97 Name: ID.NO: PLEASE, BE NEAT AND SHOW ALL YOUR WORK; CIRCLE YOUR ANSWER. NO NOTES, BOOKS, CALCULATORS, TAPE PLAYERS, or COMPUTERS. Problem
More informationf(t)e st dt. (4.1) Note that the integral defining the Laplace transform converges for s s 0 provided f(t) Ke s 0t for some constant K.
4 Laplace transforms 4. Definition and basic properties The Laplace transform is a useful tool for solving differential equations, in particular initial value problems. It also provides an example of integral
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 20D Final Exam 8 December has eigenvalues 3, 3, 0 and find the eigenvectors associated with 3. ( 2) det
Math D Final Exam 8 December 9. ( points) Show that the matrix 4 has eigenvalues 3, 3, and find the eigenvectors associated with 3. 4 λ det λ λ λ = (4 λ) det λ ( ) det + det λ = (4 λ)(( λ) 4) + ( λ + )
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 informationMath 307 Lecture 19. Laplace Transforms of Discontinuous Functions. W.R. Casper. Department of Mathematics University of Washington.
Math 307 Lecture 19 Laplace Transforms of Discontinuous Functions W.R. Casper Department of Mathematics University of Washington November 26, 2014 Today! Last time: Step Functions This time: Laplace Transforms
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 informationThe Laplace transform
The Laplace transform Samy Tindel Purdue University Differential equations - MA 266 Taken from Elementary differential equations by Boyce and DiPrima Samy T. Laplace transform Differential equations 1
More informationThird In-Class Exam Solutions Math 246, Professor David Levermore Thursday, 3 December 2009 (1) [6] Given that 2 is an eigenvalue of the matrix
Third In-Class Exam Solutions Math 26, Professor David Levermore Thursday, December 2009 ) [6] Given that 2 is an eigenvalue of the matrix A 2, 0 find all the eigenvectors of A associated with 2. Solution.
More informationCh 6.2: Solution of Initial Value Problems
Ch 6.2: Solution of Initial Value Problems! The Laplace transform is named for the French mathematician Laplace, who studied this transform in 1782.! The techniques described in this chapter were developed
More informationModes and Roots ... mx + bx + kx = 0. (2)
A solution of the form x(t) = ce rt to the homogeneous constant coefficient linear equation a x (n) + (n 1). n a n 1 x + + a 1 x + a 0 x = 0 (1) is called a modal solution and ce rt is called a mode of
More informationLaplace Transforms. Chapter 3. Pierre Simon Laplace Born: 23 March 1749 in Beaumont-en-Auge, Normandy, France Died: 5 March 1827 in Paris, France
Pierre Simon Laplace Born: 23 March 1749 in Beaumont-en-Auge, Normandy, France Died: 5 March 1827 in Paris, France Laplace Transforms Dr. M. A. A. Shoukat Choudhury 1 Laplace Transforms Important analytical
More informationExam 3 Review Sheet Math 2070
The syllabus for Exam 3 is Sections 3.6, 5.1 to 5.3, 5.5, 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,
More informationMATH 251 Final Examination December 16, 2014 FORM A. Name: Student Number: Section:
MATH 2 Final Examination December 6, 204 FORM A Name: Student Number: Section: This exam has 7 questions for a total of 0 points. In order to obtain full credit for partial credit problems, all work must
More informationLaplace Transform Theory - 1
Laplace Transform Theory - 1 Existence of Laplace Transforms Before continuing our use of Laplace transforms for solving DEs, it is worth digressing through a quick investigation of which functions actually
More informationCHEE 319 Tutorial 3 Solutions. 1. Using partial fraction expansions, find the causal function f whose Laplace transform. F (s) F (s) = C 1 s + C 2
CHEE 39 Tutorial 3 Solutions. Using partial fraction expansions, find the causal function f whose Laplace transform is given by: F (s) 0 f(t)e st dt (.) F (s) = s(s+) ; Solution: Note that the polynomial
More informationFinal Exam Sample Problems, Math 246, Spring 2018
Final Exam Sample Problems, Math 246, Spring 2018 1) Consider the differential equation dy dt = 9 y2 )y 2. a) Find all of its stationary points and classify their stability. b) Sketch its phase-line portrait
More informationChapter 6 The Laplace Transform
Ordinary Differential Equations (Math 2302) 2017-2016 Chapter 6 The Laplace Transform Many practical engineering problems involve mechanical or electrical systems acted on by discontinuous or impulsive
More information3.5 Undetermined Coefficients
3.5. UNDETERMINED COEFFICIENTS 153 11. t 2 y + ty + 4y = 0, y(1) = 3, y (1) = 4 12. t 2 y 4ty + 6y = 0, y(0) = 1, y (0) = 1 3.5 Undetermined Coefficients In this section and the next we consider the nonhomogeneous
More informationMath 341 Fall 2008 Friday December 12
FINAL EXAM: Differential Equations Math 341 Fall 2008 Friday December 12 c 2008 Ron Buckmire 1:00pm-4:00pm Name: Directions: Read all problems first before answering any of them. There are 17 pages in
More informationMathQuest: Differential Equations
MathQuest: Differential Equations Laplace Tranforms 1. True or False The Laplace transform method is the only way to solve some types of differential equations. (a) True, and I am very confident (b) True,
More informationChemical Engineering 436 Laplace Transforms (1)
Chemical Engineering 436 Laplace Transforms () Why Laplace Transforms?? ) Converts differential equations to algebraic equations- facilitates combination of multiple components in a system to get the total
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 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 informationLaplace Theory Examples
Laplace Theory Examples Harmonic oscillator s-differentiation Rule First shifting rule Trigonometric formulas Exponentials Hyperbolic functions s-differentiation Rule First Shifting Rule I and II Damped
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 informationREVIEW FOR MT3 ANSWER KEY MATH 2373, SPRING 2015
REVIEW FOR MT3 ANSWER KEY MATH 373 SPRING 15 PROF. YOICHIRO MORI This list of problems is not guaranteed to be an absolutel complete review. For completeness ou must also make sure that ou know how to
More informationApplied Differential Equation. October 22, 2012
Applied Differential Equation October 22, 22 Contents 3 Second Order Linear Equations 2 3. Second Order linear homogeneous equations with constant coefficients.......... 4 3.2 Solutions of Linear Homogeneous
More informationMATH 251 Final Examination August 14, 2015 FORM A. Name: Student Number: Section:
MATH 251 Final Examination August 14, 2015 FORM A Name: Student Number: Section: This exam has 11 questions for a total of 150 points. Show all your work! In order to obtain full credit for partial credit
More informationControl Systems. Laplace domain analysis
Control Systems Laplace domain analysis L. Lanari outline introduce the Laplace unilateral transform define its properties show its advantages in turning ODEs to algebraic equations define an Input/Output
More information27. The pole diagram and the Laplace transform
124 27. The pole diagram and the Laplace transform When working with the Laplace transform, it is best to think of the variable s in F (s) as ranging over the complex numbers. In the first section below
More informationMa 221 Final Exam Solutions 5/14/13
Ma 221 Final Exam Solutions 5/14/13 1. Solve (a) (8 pts) Solution: The equation is separable. dy dx exy y 1 y0 0 y 1e y dy e x dx y 1e y dy e x dx ye y e y dy e x dx ye y e y e y e x c The last step comes
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 informationMa 221 Final Exam 18 May 2015
Ma 221 Final Exam 18 May 2015 Print Name: Lecture Section: Lecturer This exam consists of 7 problems. You are to solve all of these problems. The point value of each problem is indicated. The total number
More informationDifferential Equations, Math 315 Midterm 2 Solutions
Name: Section: Differential Equations, Math 35 Midterm 2 Solutions. A mass of 5 kg stretches a spring 0. m (meters). The mass is acted on by an external force of 0 sin(t/2)n (newtons) and moves in a medium
More information{ sin(t), t [0, sketch the graph of this f(t) = L 1 {F(p)}.
EM Solved roblems Lalace & Fourier transform c Habala 3 EM Solved roblems Lalace & Fourier transform Find the Lalace transform of the following functions: ft t sint; ft e 3t cos3t; 3 ft e 3s ds; { sint,
More informationMath 3313: Differential Equations Laplace transforms
Math 3313: Differential Equations Laplace transforms Thomas W. Carr Department of Mathematics Southern Methodist University Dallas, TX Outline Introduction Inverse Laplace transform Solving ODEs with Laplace
More informationMATH 251 Examination II July 28, Name: Student Number: Section:
MATH 251 Examination II July 28, 2008 Name: Student Number: Section: This exam has 9 questions for a total of 100 points. In order to obtain full credit for partial credit problems, all work must be shown.
More informationUnit 2: Modeling in the Frequency Domain Part 2: The Laplace Transform. The Laplace Transform. The need for Laplace
Unit : Modeling in the Frequency Domain Part : Engineering 81: Control Systems I Faculty of Engineering & Applied Science Memorial University of Newfoundland January 1, 010 1 Pair Table Unit, Part : Unit,
More informationMA 266 FINAL EXAM INSTRUCTIONS May 8, 2010
MA 266 FINAL EXAM INSTRUCTIONS May 8, 200 NAME INSTRUCTOR. You must use a #2 pencil on the mark sense sheet (answer sheet). 2. On the mark-sense sheet, fill in the instructor s name (if you do not know,
More informationDifferential Equations Practice: 2nd Order Linear: Nonhomogeneous Equations: Variation of Parameters Page 1
Differential Equations Practice: 2nd Order Linear: Nonhomogeneous Equations: Variation of Parameters Page Questions Example (3.6.) Find a particular solution of the differential equation y 5y + 6y = 2e
More informationENGIN 211, Engineering Math. Laplace Transforms
ENGIN 211, Engineering Math Laplace Transforms 1 Why Laplace Transform? Laplace transform converts a function in the time domain to its frequency domain. It is a powerful, systematic method in solving
More informationChapter 6: The Laplace Transform 6.3 Step Functions and
Chapter 6: The Laplace Transform 6.3 Step Functions and Dirac δ 2 April 2018 Step Function Definition: Suppose c is a fixed real number. The unit step function u c is defined as follows: u c (t) = { 0
More informationOrdinary Differential Equations. Session 7
Ordinary Differential Equations. Session 7 Dr. Marco A Roque Sol 11/16/2018 Laplace Transform Among the tools that are very useful for solving linear differential equations are integral transforms. An
More information20.6. Transfer Functions. Introduction. Prerequisites. Learning Outcomes
Transfer Functions 2.6 Introduction In this Section we introduce the concept of a transfer function and then use this to obtain a Laplace transform model of a linear engineering system. (A linear engineering
More informationSolutions to Homework 3
Solutions to Homework 3 Section 3.4, Repeated Roots; Reduction of Order Q 1). Find the general solution to 2y + y = 0. Answer: The charactertic equation : r 2 2r + 1 = 0, solving it we get r = 1 as a repeated
More informationPurdue University Study Guide for MA Credit Exam
Purdue University Study Guide for MA 60 Credit Exam Students who pass the credit exam will gain credit in MA60. The credit exam is a twohour long exam with 5 multiple choice questions. No books or notes
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 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 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 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 informationLecture 7: Laplace Transform and Its Applications Dr.-Ing. Sudchai Boonto
Dr-Ing Sudchai Boonto Department of Control System and Instrumentation Engineering King Mongkut s Unniversity of Technology Thonburi Thailand Outline Motivation The Laplace Transform The Laplace Transform
More informationMath Shifting theorems
Math 37 - Shifting theorems Erik Kjær Pedersen November 29, 2005 Let us recall the Dirac delta function. It is a function δ(t) which is 0 everywhere but at t = 0 it is so large that b a (δ(t)dt = when
More informationMath 216 Final Exam 14 December, 2012
Math 216 Final Exam 14 December, 2012 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 information9.5 The Transfer Function
Lecture Notes on Control Systems/D. Ghose/2012 0 9.5 The Transfer Function Consider the n-th order linear, time-invariant dynamical system. dy a 0 y + a 1 dt + a d 2 y 2 dt + + a d n y 2 n dt b du 0u +
More informationFinal Exam December 20, 2011
Final Exam December 20, 2011 Math 420 - Ordinary Differential Equations No credit will be given for answers without mathematical or logical justification. Simplify answers as much as possible. Leave solutions
More information