Lecture Notes for Math 251: ODE and PDE. Lecture 22: 6.5 Dirac Delta and Laplace Transforms
|
|
- Elwin Brooks
- 5 years ago
- Views:
Transcription
1 Lecture Notes for Math : ODE and PDE. Lecture : 6. Dirac Delta and Laplace Transforms Shawn D. Ryan Spring 0 Dirac Delta and the Laplace Transform Last Time: We studied differential equations with discontinuous forcing functions. Now we want to look at when that forcing function is a Dirac Delta. When we considered step functions we considered these to be like switches which get turned on after a given amount of time. In applications this represented a new external force applied at a certain time. What if instead we wanted to apply a large force over a short period of time? In applications this represents a hammer striking an object once or a force applied only at one instant. We want to introduce the Dirac Delta function to accomplish this goal.. The Dirac Delta Paul Dirac (90-98) was a British Physicist who studied quantum Mechanics. Famous Quote on Poetry: In Science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in poetry, it s exactly the opposite. There are several ways to define the Dirac delta, but we will define it so it satisfies the following properties: Definition. (Dirac Delta) The Dirac delta at t = c, denoted δ(t c), satisfies the following: () δ(t c) = 0, when t c () c+ɛ δ(t c)dt =, for any ɛ > 0 c ɛ (3) c+ɛ f(t)δ(t c)dt = f(c), for any ɛ > 0. c ɛ We can think of δ(t c) as having an infinite value at t = c, so that its total energy is, all concentrated at a single point. So think of the Dirac delta as an instantaneous impulse at time t = c. The second and third properties will not work when the limits are the endpoints of any interval including t = c. This function is zero everywhere but one point, and yet the integral is. This is why it is a function, the Dirac delta is not really a function, but in higher mathematics it is referred to as a generalized function or a distribution.
2 . Laplace Transform of the Dirac Delta Before we try solving IVPs with Dirac Deltas, we will need to know its Laplace Transform. By definition L{δ(t c)} = 0 e st δ(t c)dt = e cs () by the third property of the Dirac delta. Notice that this requires c > 0 or the integral would just vanish. Now let s try solving some IVPs involving Dirac deltas Example. Solve the following initial value problem y + 3y 0y = δ(t ), y(0) =, y (0) = 3 () We begin by taking the Laplace Transform of both sides of the equation. So s Y (s) sy(0) y (0) + 3(sY (s) y(0)) 0 e s (3) By partial fractions decomposition we have (s + 3s 0)Y (s) s 3 = e s () e s (s + )(s ) + s + 3 () (s + )(s ) = Y (s)e s + Y (s) (6) Y (s) = Y (s) = (s + )(s ) = 7 s 7 s + s + 3 (s + )(s ) = s + s + (7) (8) Take inverse Laplace Transforms to get y (t) = 7 et 7 e t (9) y (t) = e t + e t (0) and the solution is then y(t) = y (t )u (t) + y (t) () ( = u (t) 7 e(t ) ) 7 e (t ) + e t + e t () ( = u (t) 7 et ) 7 e t 0 + e t + e t (3)
3 Notice that even though the exponential in the transform Y (s) came originally from the delta, once we inverse the transform the corresponding term becomes a step function. This is generally the case, because there is a relationship between the step function u c (t) and the delta δ(t c). We begin with the integral { t 0, t < c δ(u c)du = (), t > c = u c (t) () The Fundamental Theorem of Calculus says u c(t) = d ( t ) δ(u c)du = δ(t c) (6) dt Thus the Dirac delta at t = c is the derivative of the step function at t = c, which we can think of geometrically by remembering that the graph of u c (t) is horizonatal at every t c, hence at those points u c (t) = 0 and it has a jump of one at t = c. Example 3. Solve the following initial value problem. y + y + 9y = δ(t ) + e t, y(0) = 0, y (0) = (7) First we Laplace Transform both sides and solve for Y (s). Thus s Y (s) sy(0) y (0) + (sy (s) y(0)) + 9 e s + s (s + s + 9)Y (s) + = e s + s (8) (9) e s s + s (s )(s + s + 9) (0) s + s + 9 = Y (s)e s + Y (s) = Y 3 (s) () Next, we have to prepare Y (s) for the inverse transform. This will require completing the square for Y (s) and Y 3 (s), while we will need to first use partial fractions decomposition on Y (s). I leave the details to you to verify, so we now have everything in the correct form for the 3
4 inverse transform Y (s) = So their inverse transforms are s + s + 9 = (s + ) + = (s + ) + Y = (s )(s + s + 9) = ( ) s s + (s + ) + = ( ) (s + ) + s (s + ) + = ( ) s s + (s + ) + 3 (s + ) + = ( s s + (s + ) + 3 ) (s + ) + Y 3 (s) = (s + s + 9) = (s + ) + = (s + ) + y (t) = 3 t sin( t) () y (t) = (et e t cos( t) 3 e t sin( t)) (3) y 3 (t) = e t sin( t). () Thus, since our original transformed function was we obtain Y (s)e s + Y (s) Y 3 (s) () y(t) = u (t)y (t ) + y (t) y 3 (t) (6) ( = u (t) e t+ sin( t ) ) (7) + ( e t e t cos( t) 3 e t sin( ) t) (8) e t sin( t). (9)
5 Example. Solve the following initial value problem. y + 6y = u 3 (t) + δ(t ), y(0) =, y (0) = (30) Again, we begin by taking the Laplace Transform of the entire equation and applying our initial conditions, we get Solving for Y (s) s Y (s) sy(0) y (0) + 6 e 3s s (s + 6)Y (s) s = e 3s s + e s (3) + e s (3) e 3s s(s + 6) + e s s s + (33) s + 6 = Y (s)e 3s + Y (s)e s + Y 3 (s) (3) The only one of these we need to use partial fractions on is the first one. The rest can be dealt with directly, all they need is a little modification. and so the associated inverse transforms are Our solution is the inverse transform of and this will be Y (s) = s(s + 6) = 8 s s 8 s + 6 Y (s) = s + 6 = s + 6 Y 3 (s) = s + s + 6 = s s s + 6 (3) (36) (37) y (t) = 8 cos(t) (38) 8 y (t) = sin(t) (39) y 3 (t) = cos(t) + sin(t). (0) Y (s)e 3s + Y (s)e s + Y 3 (s) () y(t) = u 3 (t)y (t 3) + u (t)y (t ) + y 3 (t) () ( = u 3 (t) 8 ) cos((t 3)) + 8 u (t) sin((t )) + cos(t) + sin(t) (3) ( = u 3 (t) 8 ) cos(t ) + 8 u (t) sin(t ) + cos(t) + sin(t) () HW 6. #,,6,8,
Section 6.5 Impulse Functions
Section 6.5 Impulse Functions Key terms/ideas: Unit impulse function (technically a generalized function or distribution ) Dirac delta function Laplace transform of the Dirac delta function IVPs with forcing
More informationMath 221 Topics since the second exam
Laplace Transforms. Math 1 Topics since the second exam There is a whole different set of techniques for solving n-th order linear equations, which are based on the Laplace transform of a function. For
More informationGeneralized sources (Sect. 6.5). The Dirac delta generalized function. Definition Consider the sequence of functions for n 1, Remarks:
Generalized sources (Sect. 6.5). The Dirac delta generalized function. Definition Consider the sequence of functions for n, d n, t < δ n (t) = n, t 3 d3 d n, t > n. d t The Dirac delta generalized function
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 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 informationLecture Notes for Math 251: ODE and PDE. Lecture 7: 2.4 Differences Between Linear and Nonlinear Equations
Lecture Notes for Math 51: ODE and PDE. Lecture 7:.4 Differences Between Linear and Nonlinear Equations Shawn D. Ryan Spring 01 1 Existence and Uniqueness Last Time: We developed 1st Order ODE models for
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 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 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 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 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 informationApril 24, 2012 (Tue) Lecture 19: Impulse Function and its Laplace Transform ( 6.5)
Lecture 19: Impulse Function and its Laplace Transform ( 6.5) April 24, 2012 (Tue) Impulse Phenomena of an impulsive nature, such as the action of very large forces (or voltages) over very short intervals
More informationSection 2.1 (First Order) Linear DEs; Method of Integrating Factors. General first order linear DEs Standard Form; y'(t) + p(t) y = g(t)
Section 2.1 (First Order) Linear DEs; Method of Integrating Factors Key Terms/Ideas: General first order linear DEs Standard Form; y'(t) + p(t) y = g(t) Integrating factor; a function μ(t) that transforms
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 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 information(f g)(t) = Example 4.5.1: Find f g the convolution of the functions f(t) = e t and g(t) = sin(t). Solution: The definition of convolution is,
.5. Convolutions and Solutions Solutions of initial value problems for linear nonhomogeneous differential equations can be decomposed in a nice way. The part of the solution coming from the initial data
More informationMath Laplace transform
1 Math 371 - Laplace transform Erik Kjær Pedersen November 22, 2005 The functions we have treated with the power series method are called analytical functions y = a k t k The differential equations we
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 312 Section 7.1: Definition of a Laplace Transform
MATH 312 Section 7.1: Definition of a Laplace Transform Prof. Jonathan Duncan Walla Walla University Spring Quarter, 2008 Outline 1 The Laplace Transform 2 The Theory of Laplace Transforms 3 Conclusions
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 informationThe integrating factor method (Sect. 1.1)
The integrating factor method (Sect. 1.1) Overview of differential equations. Linear Ordinary Differential Equations. The integrating factor method. Constant coefficients. The Initial Value Problem. Overview
More information20.3. Further Laplace Transforms. Introduction. Prerequisites. Learning Outcomes
Further Laplace Transforms 2.3 Introduction In this Section we introduce the second shift theorem which simplifies the determination of Laplace and inverse Laplace transforms in some complicated cases.
More information2.161 Signal Processing: Continuous and Discrete Fall 2008
MIT OpenCourseWare http://ocw.mit.edu.6 Signal Processing: Continuous and Discrete Fall 008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS
More informationSecond order Unit Impulse Response. 1. Effect of a Unit Impulse on a Second order System
Effect of a Unit Impulse on a Second order System We consider a second order system mx + bx + kx = f (t) () Our first task is to derive the following If the input f (t) is an impulse cδ(t a), then the
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 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 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 informationLecture Notes for Math 251: ODE and PDE. Lecture 6: 2.3 Modeling With First Order Equations
Lecture Notes for Math 251: ODE and PDE. Lecture 6: 2.3 Modeling With First Order Equations Shawn D. Ryan Spring 2012 1 Modeling With First Order Equations Last Time: We solved separable ODEs and now we
More informationMAT 275 Laboratory 7 Laplace Transform and the Symbolic Math Toolbox
Laplace Transform and the Symbolic Math Toolbox 1 MAT 275 Laboratory 7 Laplace Transform and the Symbolic Math Toolbox In this laboratory session we will learn how to 1. Use the Symbolic Math Toolbox 2.
More information2t t dt.. So the distance is (t2 +6) 3/2
Math 8, Solutions to Review for the Final Exam Question : The distance is 5 t t + dt To work that out, integrate by parts with u t +, so that t dt du The integral is t t + dt u du u 3/ (t +) 3/ So the
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS Basic Concepts Paul Dawkins Table of Contents Preface... Basic Concepts... 1 Introduction... 1 Definitions... Direction Fields... 8 Final Thoughts...19 007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx
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 informationChapter 6: The Definite Integral
Name: Date: Period: AP Calc AB Mr. Mellina Chapter 6: The Definite Integral v v Sections: v 6.1 Estimating with Finite Sums v 6.5 Trapezoidal Rule v 6.2 Definite Integrals 6.3 Definite Integrals and Antiderivatives
More informationMATH 113: ELEMENTARY CALCULUS
MATH 3: ELEMENTARY CALCULUS Please check www.ualberta.ca/ zhiyongz for notes updation! 6. Rates of Change and Limits A fundamental philosophical truth is that everything changes. In physics, the change
More information= lim. (1 + h) 1 = lim. = lim. = lim = 1 2. lim
Math 50 Exam # Solutions. Evaluate the following its or explain why they don t exist. (a) + h. h 0 h Answer: Notice that both the numerator and the denominator are going to zero, so we need to think a
More information[1] Model of on/off process: a light turns on; first it is dark, then it is light. The basic model is the Heaviside unit step function
18.03 Class 23, April 2, 2010 Step and delta [1] Step function u(t) [2] Rates and delta(t) [3] Regular, singular, and generalized functions [4] Generalized derivative [5] Heaviside and Dirac Two additions
More informationLecture 3 Partial Differential Equations
Lecture 3 Partial Differential Equations Prof. Massimo Guidolin Prep Course in Investments August-September 2016 Plan of the lecture Motivation and generalities The heat equation and its applications in
More informationORDINARY DIFFERENTIAL EQUATIONS
ORDINARY DIFFERENTIAL EQUATIONS GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI, 4884 NOVEMBER 9, 7 Summary This is an introduction to ordinary differential equations We
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 informationORDINARY DIFFERENTIAL EQUATIONS
ORDINARY DIFFERENTIAL EQUATIONS GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI, 48824. JANUARY 3, 25 Summary. This is an introduction to ordinary differential equations.
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 106 Calculus 1 Topics for first exam
Chapter 2: Limits and Continuity Rates of change and its: Math 06 Calculus Topics for first exam Limit of a function f at a point a = the value the function should take at the point = the value that the
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 informationBasics Quantum Mechanics Prof. Ajoy Ghatak Department of physics Indian Institute of Technology, Delhi
Basics Quantum Mechanics Prof. Ajoy Ghatak Department of physics Indian Institute of Technology, Delhi Module No. # 03 Linear Harmonic Oscillator-1 Lecture No. # 04 Linear Harmonic Oscillator (Contd.)
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 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 information17. Impulses and generalized functions
86 17. Impulses and generalized functions In calculus you learn how to model processes using functions. Functions have their limitations, though, and, at least as they are treated in calculus, they are
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 informationLecture 1 Signals. notation and meaning. common signals. size of a signal. qualitative properties of signals. impulsive signals
EE102 spring 2002 Handout #2 Lecture 1 Signals notation and meaning common signals size of a signal qualitative properties of signals impulsive signals 1 1 Signals a signal is a function of time, e.g.,
More informationShort Solutions to Review Material for Test #2 MATH 3200
Short Solutions to Review Material for Test # MATH 300 Kawai # Newtonian mechanics. Air resistance. a A projectile is launched vertically. Its height is y t, and y 0 = 0 and v 0 = v 0 > 0. The acceleration
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 information2.1 Differential Equations and Solutions. Blerina Xhabli
2.1 Math 3331 Differential Equations 2.1 Differential Equations and Solutions Blerina Xhabli Department of Mathematics, University of Houston blerina@math.uh.edu math.uh.edu/ blerina/teaching.html Blerina
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 informationTime Response of Systems
Chapter 0 Time Response of Systems 0. Some Standard Time Responses Let us try to get some impulse time responses just by inspection: Poles F (s) f(t) s-plane Time response p =0 s p =0,p 2 =0 s 2 t p =
More informationMITOCW watch?v=poho4pztw78
MITOCW watch?v=poho4pztw78 GILBERT STRANG: OK. So this is a video in which we go for second-order equations, constant coefficients. We look for the impulse response, the key function in this whole business,
More information2tdt 1 y = t2 + C y = which implies C = 1 and the solution is y = 1
Lectures - Week 11 General First Order ODEs & Numerical Methods for IVPs In general, nonlinear problems are much more difficult to solve than linear ones. Unfortunately many phenomena exhibit nonlinear
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 informationBasic Procedures for Common Problems
Basic Procedures for Common Problems ECHE 550, Fall 2002 Steady State Multivariable Modeling and Control 1 Determine what variables are available to manipulate (inputs, u) and what variables are available
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 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 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 informatione (x y)2 /4kt φ(y) dy, for t > 0. (4)
Math 24A October 26, 2 Viktor Grigoryan Heat equation: interpretation of the solution Last time we considered the IVP for the heat equation on the whole line { ut ku xx = ( < x
More information18.01 Single Variable Calculus Fall 2006
MIT OpenCourseWare http://ocw.mit.edu 8.0 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Lecture 0 8.0 Fall 2006 Lecture
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 information2. A Motivational Example: Consider the second-order homogeneous linear DE: y y 2y = 0
MATH 246: Chapter 2 Section 2 Justin Wyss-Gallifent 1. Introduction: Since even linear higher-order DEs are difficult we are going to simplify even more. For today we re going to look at homogeneous higher-order
More information2.2 Average vs. Instantaneous Description
2 KINEMATICS 2.2 Average vs. Instantaneous Description Name: 2.2 Average vs. Instantaneous Description 2.2.1 Average vs. Instantaneous Velocity In the previous activity, you figured out that you can calculate
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 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 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 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 information6.003 Homework #10 Solutions
6.3 Homework # Solutions Problems. DT Fourier Series Determine the Fourier Series coefficients for each of the following DT signals, which are periodic in N = 8. x [n] / n x [n] n x 3 [n] n x 4 [n] / n
More informationName: Solutions Exam 3
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, 5.5, 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 information226 My God, He Plays Dice! Entanglement. Chapter This chapter on the web informationphilosopher.com/problems/entanglement
226 My God, He Plays Dice! Entanglement Chapter 29 20 This chapter on the web informationphilosopher.com/problems/entanglement Entanglement 227 Entanglement Entanglement is a mysterious quantum phenomenon
More informationFINAL EXAM SOLUTIONS, MATH 123
FINAL EXAM SOLUTIONS, MATH 23. Find the eigenvalues of the matrix ( 9 4 3 ) So λ = or 6. = λ 9 4 3 λ = ( λ)( 3 λ) + 36 = λ 2 7λ + 6 = (λ 6)(λ ) 2. Compute the matrix inverse: ( ) 3 3 = 3 4 ( 4/3 ) 3. Let
More informationAnalysis III for D-BAUG, Fall 2017 Lecture 10
Analysis III for D-BAUG, Fall 27 Lecture Lecturer: Alex Sisto (sisto@math.ethz.ch Convolution (Faltung We have already seen that the Laplace transform is not multiplicative, that is, L {f(tg(t} L {f(t}
More informationLecture 29. Convolution Integrals and Their Applications
Math 245 - Mathematics of Physics and Engineering I Lecture 29. Convolution Integrals and Their Applications March 3, 212 Konstantin Zuev (USC) Math 245, Lecture 29 March 3, 212 1 / 13 Agenda Convolution
More informationORDINARY DIFFERENTIAL EQUATIONS
Page 1 of 15 ORDINARY DIFFERENTIAL EQUATIONS Lecture 17 Delta Functions & The Impulse Response (Revised 30 March, 2009 @ 09:40) Professor Stephen H Saperstone Department of Mathematical Sciences George
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 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 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 124A October 11, 2011
Math 14A October 11, 11 Viktor Grigoryan 6 Wave equation: solution In this lecture we will solve the wave equation on the entire real line x R. This corresponds to a string of infinite length. Although
More informationLecture 1 From Continuous-Time to Discrete-Time
Lecture From Continuous-Time to Discrete-Time Outline. Continuous and Discrete-Time Signals and Systems................. What is a signal?................................2 What is a system?.............................
More informationLecture 10. (2) Functions of two variables. Partial derivatives. Dan Nichols February 27, 2018
Lecture 10 Partial derivatives Dan Nichols nichols@math.umass.edu MATH 233, Spring 2018 University of Massachusetts February 27, 2018 Last time: functions of two variables f(x, y) x and y are the independent
More information1 4 r q. (9.172) G(r, q) =
FOURIER ANALYSIS: LECTURE 5 9 Green s functions 9. Response to an impulse We have spent some time so far in applying Fourier methods to solution of di erential equations such as the damped oscillator.
More informationMATH 353 LECTURE NOTES: WEEK 1 FIRST ORDER ODES
MATH 353 LECTURE NOTES: WEEK 1 FIRST ORDER ODES J. WONG (FALL 2017) What did we cover this week? Basic definitions: DEs, linear operators, homogeneous (linear) ODEs. Solution techniques for some classes
More informationEQ: How do I convert between standard form and scientific notation?
EQ: How do I convert between standard form and scientific notation? HW: Practice Sheet Bellwork: Simplify each expression 1. (5x 3 ) 4 2. 5(x 3 ) 4 3. 5(x 3 ) 4 20x 8 Simplify and leave in standard form
More informationLecture Notes for Math 251: ODE and PDE. Lecture 30: 10.1 Two-Point Boundary Value Problems
Lecture Notes for Math 251: ODE and PDE. Lecture 30: 10.1 Two-Point Boundary Value Problems Shawn D. Ryan Spring 2012 Last Time: We finished Chapter 9: Nonlinear Differential Equations and Stability. Now
More informationFrequency Response (III) Lecture 26:
EECS 20 N March 21, 2001 Lecture 26: Frequency Response (III) Laurent El Ghaoui 1 outline reading assignment: Chapter 8 of Lee and Varaiya we ll concentrate on continuous-time systems: convolution integral
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 informationEA2.3 - Electronics 2 1
In the previous lecture, I talked about the idea of complex frequency s, where s = σ + jω. Using such concept of complex frequency allows us to analyse signals and systems with better generality. In this
More informationLaplace Transform Introduction
Laplace Transform Introduction In many problems, a function is transformed to another function through a relation of the type: where is a known function. Here, is called integral transform of. Thus, an
More information9.2 The Input-Output Description of a System
Lecture Notes on Control Systems/D. Ghose/212 22 9.2 The Input-Output Description of a System The input-output description of a system can be obtained by first defining a delta function, the representation
More informationCHAPTER 6 VECTOR CALCULUS. We ve spent a lot of time so far just looking at all the different ways you can graph
CHAPTER 6 VECTOR CALCULUS We ve spent a lot of time so far just looking at all the different ways you can graph things and describe things in three dimensions, and it certainly seems like there is a lot
More informationChapter 1: Introduction
Chapter 1: Introduction Definition: A differential equation is an equation involving the derivative of a function. If the function depends on a single variable, then only ordinary derivatives appear and
More informationPlane Curves and Parametric Equations
Plane Curves and Parametric Equations MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction We typically think of a graph as a curve in the xy-plane generated by the
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 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 informationSection 2.1 Differential Equation and Solutions
Section 2.1 Differential Equation and Solutions Key Terms: Ordinary Differential Equation (ODE) Independent Variable Order of a DE Partial Differential Equation (PDE) Normal Form Solution General Solution
More informationContinuity and One-Sided Limits
Continuity and One-Sided Limits 1. Welcome to continuity and one-sided limits. My name is Tuesday Johnson and I m a lecturer at the University of Texas El Paso. 2. With each lecture I present, I will start
More information