The Laplace Transform and the IVP (Sect. 6.2).
|
|
|
- Shana Chambers
- 6 years ago
- Views:
Transcription
1 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. Solving differential equations using L ]. Remark: The method works with: Constant coefficient equations. Homogeneous and non-homogeneous equations. First, second, higher order equations. Idea of the method: ] differential eq. L for y(t). () Algebraic eq. for Ly(t)]. (2) (2) Solve the algebraic eq. for Ly(t)]. (3) Transform back to obtain y(t). (Using the table.)
2 Solving differential equations using L ]. Idea of the method: ] differential eq. L for y(t). (2) Solve the algebraic eq. for Ly(t)]. () (3) Algebraic eq. for Ly(t)]. Transform back to obtain y(t). (2) (Using the table.) Recall: (a) La f (t) + b g(t)] = a Lf (t)] + b Lg(t)]; (b) L y (n)] = s n Ly] s (n ) y(0) s (n 2) y (0) y (n ) (0). 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.
3 Homogeneous IVP. y y 2y = 0, y(0) =, y (0) = 0. Solution: Compute the L ] of the differential equation, Ly y 2y] = L0] Ly y 2y] = 0. The L ] is a linear function, so Ly ] Ly ] 2 Ly] = 0. Derivatives are transformed into power functions, ] ] s 2 Ly] s y(0) y (0) s Ly] y(0) 2 Ly] = 0, We the obtain (s 2 s 2) Ly] = (s ) y(0) + y (0). Homogeneous IVP. y y 2y = 0, y(0) =, y (0) = 0. Solution: Recall: (s 2 s 2) Ly] = (s ) y(0) + y (0). Differential equation for y Introduce the initial condition, L ] Algebraic equation for Ly]. (s 2 s 2) Ly] = (s ). We can solve for the unknown Ly] as follows, Ly] = (s ) (s 2 s 2).
4 Homogeneous IVP. y y 2y = 0, y(0) =, y (0) = 0. Solution: Recall: Ly] = (s ) (s 2 s 2). The partial fraction method: Find the zeros of the denominator, { s 2 s 2 = 0 s ± = ] s+ = 2, ± s =, Therefore, we rewrite: Ly] = Find constants a and b such that (s ) (s 2)(s + ). (s ) (s 2)(s + ) = a s 2 + b s +. Homogeneous IVP. Solution: Recall: y y 2y = 0, y(0) =, y (0) = 0. A simple calculation shows (s ) (s 2)(s + ) = a s 2 + (s ) (s 2)(s + ) = a s 2 + b s + b s +. a(s + ) + b(s 2) = (s 2)(s + ) (s ) = s(a + b) + (a 2b) { a + b =, a 2b = Hence, a = 3 and b = 2 3. Then, Ly] = 3 (s 2) (s + ).
5 Homogeneous IVP. y y 2y = 0, y(0) =, y (0) = 0. Solution: Recall: Ly] = 3 (s 2) From the table: (s + ) Le at ] = s a s 2 = Le2t ], s + = Le t ]. So we arrive at the equation Ly] = 3 Le2t ] Le t ] = L e 3( 2t + 2e t)] We conclude that: y(t) = 3( e 2t + 2e t). Homogeneous IVP. y 4y + 4y = 0, y(0) =, y (0) =. Solution: Compute the L ] of the differential equation, The L ] is a linear function, Ly 4y + 4y] = L0] = 0. Ly ] 4 Ly ] + 4 Ly] = 0. Derivatives are transformed into power functions, ] ] s 2 Ly] s y(0) y (0) 4 s Ly] y(0) + 4 Ly] = 0, Therefore, (s 2 4s + 4) Ly] = (s 4) y(0) + y (0).
6 Homogeneous IVP. y 4y + 4y = 0, y(0) =, y (0) =. Solution: Recall: (s 2 4s + 4) Ly] = (s 4) y(0) + y (0). Introduce the initial conditions, (s 2 4s + 4) Ly] = s 3. Solve for Ly] as follows: Ly] = (s 3) (s 2 4s + 4). The partial fraction method: Find the roots of the denominator, s 2 4s +4 = 0 s ± = 2 We obtain: Ly] = (s 3) (s 2) 2. 4± ] s + = s = 2. Homogeneous IVP. y 4y + 4y = 0, y(0) =, y (0) =. Solution: Recall: Ly] = (s 3) (s 2) 2. This expression is already in the partial fraction decomposition. Idea: Rewrite the right-hand side in terms of function in the table. Ly] = (s 2) (s 2) 2 = From the Laplace transforms table: Le at ] = s a Lt n e at n! ] = (s a) (n+) (s 2) (s 2) 2 (s 2) 2 = s 2 (s 2) 2. s 2 = Le2t ], (s 2) 2 = Lte2t ].
7 Homogeneous IVP. y 4y + 4y = 0, y(0) =, y (0) =. Solution: Recall: Ly] = s 2 (s 2) 2 and s 2 = Le2t ], (s 2) 2 = Lte2t ]. So we arrive at the equation Ly] = Le 2t ] Lte 2t ] = L e 2t te 2t]. We conclude that y(t) = e 2t te 2t. 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.
8 First, second, higher order equations. Use the Laplace Transform to find the solution of y (4) 4y = 0, y(0) =, y (0) =, y (0) = 2, y (0) = 0. Solution: Compute the L ] of the equation, L y (4)] 4 Ly] = 0. s 4 Ly] s 3 y(0) s 2 y (0) s y (0) y (0) ] 4 Ly] = 0. s 4 Ly] s 3 + 2s ] 4 Ly] = 0 (s 4 4) Ly] = s 3 2s, We obtain, Ly] = s3 2s (s 4 4). First, second, higher order equations. Use the Laplace Transform to find the solution of y (4) 4y = 0, y(0) =, y (0) =, y (0) = 2, y (0) = 0. Solution: Recall: Ly] = Ly] = s3 2s (s 4 4). s(s 2 2) (s 2 2)(s 2 + 2) Ly] = s (s 2 + 2). The last expression is in the table of Laplace Transforms, Ly] = s ( s ] 2) = L cos( 2 t) ]. We conclude that y(t) = cos( 2 t).
9 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. Non-homogeneous IVP. y 4y + 4y = 3 sin(2t), y(0) =, y (0) =. Solution: Compute the Laplace transform of the equation, Ly 4y + 4y] = L3 sin(2t)]. The right-hand side above can be expressed as follows, L3 sin(2t)] = 3 Lsin(2t)] = 3 2 s = Introduce this source term in the differential equation, Ly ] 4 Ly ] + 4 Ly] =
10 Non-homogeneous IVP. y 4y + 4y = 3 sin(2t), y(0) =, y (0) =. Solution: Recall: Ly ] 4 Ly ] + 4 Ly] = Derivatives are transformed into power functions, s 2 Ly] s y(0) y (0) Rewrite the above equation, ] 4 s Ly] y(0) ] + 4 Ly] = (s 2 4s + 4) Ly] = (s 4) y(0) + y (0) + Introduce the initial conditions, (s 2 4s + 4) Ly] = s 3 + Non-homogeneous IVP. y 4y + 4y = 3 sin(2t), y(0) =, y (0) =. Solution: Recall: (s 2 4s + 4) Ly] = s 3 + Therefore, Ly] = (s 3) (s 2 4s + 4) + (s )(s 2 + 4). From an above: s 2 4s + 4 = (s 2) 2, Ly] = s 2 (s 2) 2 + (s 2) 2 (s 2 + 4). From an above we know that Le 2t te 2t ] = s 2 (s 2) 2.
11 Non-homogeneous IVP. y 4y + 4y = 3 sin(2t), y(0) =, y (0) =. Solution: Recall: Ly] = Le 2t te 2t ] + (s 2) 2 (s 2 + 4). Use Partial fractions to simplify the last term above. Find constants a, b, c, d, such that (s 2) 2 (s 2 + 4) = as + b s c (s 2) + d (s 2) 2 (s 2) 2 (s 2 + 4) = (as + b)(s 2)2 + c(s 2)(s 2 + 4) + d(s 2 + 4) (s 2 + 4)(s 2) 2 = (as + b)(s 2) 2 + c(s 2)(s 2 + 4) + d(s 2 + 4). Non-homogeneous IVP. y 4y + 4y = 3 sin(2t), y(0) =, y (0) =. Solution: = (as + b)(s 2) 2 + c(s 2)(s 2 + 4) + d(s 2 + 4). = (as + b)(s 2 4s + 4) + c(s 3 + 4s 2s 2 8) + d(s 2 + 4) = a(s 3 4s 2 +4s)+b(s 2 4s +4)+c(s 3 +4s 2s 2 8)+d(s 2 +4). We obtain the system = (a + c)s 3 + ( 4a + b 2c + d)s 2 + (4a 4b + 4c)s + (4b 8c + 4d). a + c= 0, 4a + b 2c + d= 0, 4a 4b + 4c= 0, 4b 8c + 4d =.
12 Non-homogeneous IVP. y 4y + 4y = 3 sin(2t), y(0) =, y (0) =. Solution: The solution for this linear system is a = 3 8, b = 0, c = 3 8, d = 3 4. (s 2) 2 (s 2 + 4) = 3 8 s s Use the table of Laplace Transforms (s 2) (s 2) 2. (s 2) 2 (s 2 + 4) = 3 8 Lcos(2t)] 3 8 Le2t ] Lte2t ]. (s 2) 2 (s 2 + 4) = L 3 8 cos(2t) 3 8 e2t te2t]. Non-homogeneous IVP. y 4y + 4y = 3 sin(2t), y(0) =, y (0) =. Solution: Summary: Ly] = Le 2t te 2t ] + (s 2) 2 (s 2 + 4), (s 2) 2 (s 2 + 4) = L 3 8 cos(2t) 3 8 e2t te2t]. Ly(t)] = L ( t) e 2t ( + 2t) e2t cos(2t) ]. We conclude that y(t) = ( t) e 2t (2t ) e2t cos(2t).
Computing 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
(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
The Laplace Transform (Sect. 4.1). The Laplace Transform (Sect. 4.1).
The Laplace Transform (Sect. 4.1). s of Laplace Transforms. The Laplace Transform (Sect. 4.1). s of Laplace Transforms. The definition of the Laplace Transform. Definition The function F : D F R is the
Find 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
Chapter 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
Laplace 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
Ch 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
Laplace 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
MathQuest: 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,
Math 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
Sample Quiz 8, Problem 1. Solving Higher Order Constant-Coefficient Equations
Sample Quiz 8 Sample Quiz 8, Problem. Solving Higher Order Constant-Coefficient Equations The Algorithm applies to constant-coefficient homogeneous linear differential equations of order N, for example
Math 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..
APPM 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
Math 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
Math 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
Math 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
f(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
Math 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
Generalized 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
The 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
MATH 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
Math 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
Section 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
3.5 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
SOLUTIONS FOR HOMEWORK SECTION 6.4 AND 6.5
SOLUTIONS FOR HOMEWORK SECTION 6.4 AND 6.5 Problem : For each of the following function do the following: (i) Write the function a a piecewie function and ketch it graph, (ii) Write the function a a combination
Name: 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
Ordinary 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
Solutions of the Sample Problems for the Third In-Class Exam Math 246, Fall 2017, Professor David Levermore
Solutions of the Sample Problems for the Third In-Class Exam Math 6 Fall 07 Professor David Levermore Compute the Laplace transform of ft t e t ut from its definition Solution The definition of the Laplace
Differential 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
Lecture 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
Solutions 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
+ + 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)
Differential Equations
Differential Equations Math 341 Fall 21 MWF 2:3-3:25pm Fowler 37 c 21 Ron Buckmire http://faculty.oxy.edu/ron/math/341/1/ Worksheet 29: Wednesday December 1 TITLE Laplace Transforms and Introduction to
Ordinary 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
Solutions 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
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
37. 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
On linear and non-linear equations.(sect. 2.4).
On linear and non-linear equations.sect. 2.4). Review: Linear differential equations. Non-linear differential equations. Properties of solutions to non-linear ODE. The Bernoulli equation. Review: Linear
JUST THE MATHS UNIT NUMBER LAPLACE TRANSFORMS 3 (Differential equations) A.J.Hobson
JUST THE MATHS UNIT NUMBER 16.3 LAPLACE TRANSFORMS 3 (Differential equations) by A.J.Hobson 16.3.1 Examples of solving differential equations 16.3.2 The general solution of a differential equation 16.3.3
Math 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
Modeling with first order equations (Sect. 2.3). The mathematical modeling of natural processes.
Modeling with first order equations (Sect. 2.3). The mathematical modeling of natural processes. Main example: Salt in a water tank. The experimental device. The main equations. Analysis of the mathematical
e 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
CHAPTER 8 Laplace Transforms
CHAPTER Laplace Transforms IN THIS CHAPTER we study the method of Laplace transforms, which illustrates one of the basic problem solving techniques in mathematics: transform a difficult problem into an
Review Sol. of More Long Answer Questions
Review Sol. of More Long Answer Questions 1. Solve the integro-differential equation t y (t) e t v y(v)dv = t; y()=. (1) Solution. The key is to recognize the convolution: t e t v y(v) dv = e t y. () Now
Third 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 246, Professor David Levermore Thursday, 3 December 2009 (1) [6] Given that 2 is an eigenvalue of the matrix](/thumbs/92/108365725.jpg "Third 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.
Math 216 Second Midterm 17 November, 2016
Math 216 Second Midterm 17 November, 2016 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
Example. Mathematics 255: Lecture 17. Example. Example (cont d) Consider the equation. d 2 y dt 2 + dy
Mathematics 255: Lecture 17 Undetermined Coefficients Dan Sloughter Furman University October 10, 2008 6y = 5e 4t. so the general solution of 0 = r 2 + r 6 = (r + 3)(r 2), 6y = 0 y(t) = c 1 e 3t + c 2
{ 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,
Chapter 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
Lecture 23 Using Laplace Transform to Solve Differential Equations
Lecture 23 Using Laplace Transform to Solve Differential Equations /02/20 Laplace transform and Its properties. Definition: L{f (s) 6 F(s) 6 0 e st f(t)dt. () Whether L{f (s) or F(s) is used depends on
MATH 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
HIGHER-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
Warm-up: What is the Laplace transform of f(t) = e^{-t} cos(3t)? We could do this by writing it as (1/2)( e^{(-1+3i)t} + e^{(-1-3i)t} )
18.03 Class 27, April 12, 2010 Laplace Transform II 1. Delta signal 2. t-derivative rule 3. Inverse transform 4. Unit impulse response 5. Partial fractions 6. L[f'_r] Laplace Transform: F(s) = int_0^\infty
HIGHER-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
4. Higher Order Linear DEs
4. Higher Order Linear DEs Department of Mathematics & Statistics ASU Outline of Chapter 4 1 General Theory of nth Order Linear Equations 2 Homogeneous Equations with Constant Coecients 3 The Method of
Math Exam 3 Solutions
Math 6 - Exam 3 Solutions Thursday, July 3rd, 0 Recast the following higher-order differential equations into first order systems If the equation is linear, be sure to give the coefficient matrix At and
Linear Variable coefficient equations (Sect. 2.1) Review: Linear constant coefficient equations
Linear Variable coefficient equations (Sect. 2.1) Review: Linear constant coefficient equations. The Initial Value Problem. Linear variable coefficients equations. The Bernoulli equation: A nonlinear equation.
Exam 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.
Solutions to Homework 5, Introduction to Differential Equations, 3450: , Dr. Montero, Spring y 4y = 48t 3.
Solutions to Homework 5, Introduction to Differential Equations, 3450:335-003, Dr. Montero, Spring 2009 Problem 1. Find a particular solution to the differential equation 4y = 48t 3. Solution: First we
ENGIN 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
Partial Fractions and the Coverup Method Haynes Miller and Jeremy Orloff
Partial Fractions and the Coverup Method 8.03 Haynes Miller and Jeremy Orloff *Much of this note is freely borrowed from an MIT 8.0 note written by Arthur Mattuck. Heaviside Cover-up Method. Introduction
On linear and non-linear equations. (Sect. 1.6).
On linear and non-linear equations. (Sect. 1.6). Review: Linear differential equations. Non-linear differential equations. The Picard-Lindelöf Theorem. Properties of solutions to non-linear ODE. The Proof
we 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
6. 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
Differential Equations Class Notes
Differential Equations Class Notes Dan Wysocki Spring 213 Contents 1 Introduction 2 2 Classification of Differential Equations 6 2.1 Linear vs. Non-Linear.................................. 7 2.2 Seperable
Honors 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
MA 201, Mathematics III, July-November 2016, Laplace Transform
MA 21, Mathematics III, July-November 216, Laplace Transform Lecture 18 Lecture 18 MA 21, PDE (216) 1 / 21 Laplace Transform Let F : [, ) R. If F(t) satisfies the following conditions: F(t) is piecewise
Ma 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
APPM 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,
Chapter 4. The Laplace Transform Method
Chapter 4. The Laplace Tranform Method The Laplace Tranform i a tranformation, meaning that it change a function into a new function. Actually, it i a linear tranformation, becaue it convert a linear combination
CHEE 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
MA 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
Section 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
Nonhomogeneous Linear Equations: Variation of Parameters Professor David Levermore 17 October 2004 We now return to the discussion of the general case
Nonhomogeneous Linear Equations: Variation of Parameters Professor David Levermore 17 October 2004 We now return to the discussion of the general case L(t)y = a 0 (t)y + a 1 (t)y + a 2 (t)y = b(t). (1.1)
A 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
Solution: Homework 3 Biomedical Signal, Systems and Control (BME )
Solution: Homework Biomedical Signal, Systems and Control (BME 80.) Instructor: René Vidal, E-mail: rvidal@cis.jhu.edu TA: Donavan Cheng, E-mail: donavan.cheng@gmail.com TA: Ertan Cetingul, E-mail: ertan@cis.jhu.edu
Non-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
20.5. The Convolution Theorem. Introduction. Prerequisites. Learning Outcomes
The Convolution Theorem 2.5 Introduction In this Section we introduce the convolution of two functions f(t), g(t) which we denote by (f g)(t). The convolution is an important construct because of the convolution
Laplace Transform Problems
AP Calculus BC Name: Laplace Transformation Day 3 2 January 206 Laplace Transform Problems Example problems using the Laplace Transform.. Solve the differential equation y! y = e t, with the initial value
Stability. X(s) Y(s) = (s + 2) 2 (s 2) System has 2 poles: points where Y(s) -> at s = +2 and s = -2. Y(s) 8X(s) G 1 G 2
Stability 8X(s) X(s) Y(s) = (s 2) 2 (s 2) System has 2 poles: points where Y(s) -> at s = 2 and s = -2 If all poles are in region where s < 0, system is stable in Fourier language s = jω G 0 - x3 x7 Y(s)
Section 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
Matemáticas II: Segundo del Grado en Ingeniería Aeroespacial
Matemáticas II: Segundo del Grado en Ingeniería Aeroespacial Sergio Blanes, Dolors Roselló Ecuaciones diferenciales y transformadas de Laplace con aplicaciones L.M. Sánchez, M.P. Legua Ref.: 211-798 Capítulo
Math 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
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
Review for Exam #3 MATH 3200
Review for Exam #3 MATH 3 Lodwick/Kawai You will have hrs. to complete Exam #3. There will be one full problem from Laplace Transform with the unit step function. There will be linear algebra, but hopefull,
Applied 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
The formulas for derivatives are particularly useful because they reduce ODEs to algebraic expressions. Consider the following ODE d 2 dx + p d
Solving ODEs using Fourier Transforms The formulas for derivatives are particularly useful because they reduce ODEs to algebraic expressions. Consider the following ODE d 2 dx + p d 2 dx + q f (x) R(x)
MATH 307: Problem Set #7
MATH 307: Problem Set #7 Due on: Feb 11, 2016 Problem 1 First-order Variation of Parameters The method of variation of parameters uses the homogeneous solutions of a linear ordinary differential equation
Short 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
Laplace Transform of Discontinuous Functions
of Dicontinuou Function Week November 7, 06 Week of Dicontinuou Function Example: If f (t) = u (t) + t (u 3(t) u 6(t)), then what i f (5)? Recall, that by definition: u (5) =, u 3(5) = and u 6(5) = 0.
Week #9 : DEs with Non-Constant Coefficients, Laplace Resonance
Week #9 : DEs with Non-Constant Coefficients, Laplace Resonance Goals: Solving DEs with Non-Constant Coefficients Resonance with Laplace Laplace with Periodic Functions 1 Solving Equations with Non-Constant
Math 333 Qualitative Results: Forced Harmonic Oscillators
Math 333 Qualitative Results: Forced Harmonic Oscillators Forced Harmonic Oscillators. Recall our derivation of the second-order linear homogeneous differential equation with constant coefficients: my
( ) f (k) = FT (R(x)) = R(k)
Solving ODEs using Fourier Transforms The formulas for derivatives are particularly useful because they reduce ODEs to algebraic expressions. Consider the following ODE d 2 dx + p d 2 dx + q f (x) = R(x)
The 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
REVIEW 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
HIGHER-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS II: Nonhomogeneous Equations. David Levermore Department of Mathematics University of Maryland
HIGHER-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS II: Nonhomogeneous Equations David Levermore Department of Mathematics University of Maryland 14 March 2012 Because the presentation of this material
Math 215/255: Elementary Differential Equations I Harish N Dixit, Department of Mathematics, UBC
Math 215/255: Elementary Differential Equations I Harish N Dixit, Department of Mathematics, UBC First Order Equations Linear Equations y + p(x)y = q(x) Write the equation in the standard form, Calculate
Mathematics for Engineers and Scientists 4 Notes for F1.8XD2
Mathematics for Engineers and Scientists 4 Notes for F.8XD2 28 This is a one-semester course building on Mathematics for Engineers and Scientists, 2 and 3. There are three main parts of the course: ( Laplace
2. First Order Linear Equations and Bernoulli s Differential Equation
August 19, 2013 2-1 2. First Order Linear Equations and Bernoulli s Differential Equation First Order Linear Equations A differential equation of the form y + p(t)y = g(t) (1) is called a first order scalar
Ch 6.4: Differential Equations with Discontinuous Forcing Functions
Ch 6.4: Differential Equations with Discontinuous Forcing Functions! In this section focus on examples of nonhomogeneous initial value problems in which the forcing function is discontinuous. Example 1: