#### Laplace

Portland, GE 222
Excerpt: ... Introduction to the Laplace Transform Mateo Aboy Lecture Outline, Examples, and Homework 1 Outline Introduction & Denition Basis Functions Functional and Operational Transforms Inverse Laplace Transform Initial and Final Value Theorems Examples Homework 2 Lecture Notes 1 3 Example Problem Use the denition to nd the functional Laplace transform of f (t) = (t) Solution: 4 Example Problem Use the denition to nd the functional Laplace transform of f (t) = u(t) Solution: 5 Example Problem Use the denition to nd the functional Laplace transform of f (t) = eat Solution: 2 6 Example Problem Use the denition to nd the functional Laplace transform of f (t) = cos(wt) Solution: 7 Example Problem Use the denition to nd the functional Laplace transform of f (t) = cos(wt) Solution: 8 Example Problem List the following operational transforms: multiplication by a constant, addition, dierentiatio ...

#### 7-2_math312

Walla Walla University, MATH 312
Excerpt: ... Inverse Laplace Transform ations Laplace Transformations and Derivatives Conclusions MATH 312 Section 7.2: Inverse Laplace Transform ations Prof. Jonathan Duncan Walla Walla College Spring Quarter, 2007 Inverse Laplace Transform ations Laplace Transformations and Derivatives Conclusions Outline 1 Inverse Laplace Transform ations 2 Laplace Transformations and Derivatives 3 Conclusions Inverse Laplace Transform ations Laplace Transformations and Derivatives Conclusions Undoing the Transformation In order to use Laplace Transformations to solve differential equations, we have to be able to undo the transformation. This is not a new concept. Derivatives and Anti-derivatives The inverse of the differential operator transformation is the indefinite integral transformation. That is, if Df (x) = g (x) then g (x) dx = f (x) (up to a constant). This same idea applies to the Laplace Transformation. Inverse Laplace Transform ations The inverse Laplace Transform ation of a function F (s) is a function f (t) s ...

#### 7-2_math312

Walla Walla University, MATH 312
Excerpt: ... Inverse Laplace Transform ations Laplace Transformations and Derivatives Conclusions MATH 312 Section 7.2: Inverse Laplace Transform ations Prof. Jonathan Duncan Walla Walla University Spring Quarter, 2008 Inverse Laplace Transform ations Laplace Transformations and Derivatives Conclusions Outline 1 Inverse Laplace Transform ations 2 Laplace Transformations and Derivatives 3 Conclusions Inverse Laplace Transform ations Laplace Transformations and Derivatives Conclusions Undoing the Transformation In order to use Laplace Transformations to solve differential equations, we have to be able to undo the transformation. This is not a new concept. Derivatives and Anti-derivatives The inverse of the differential operator transformation is the indefinite integral transformation. That is, if Df (x) = g (x) then g (x) dx = f (x) (up to a constant). This same idea applies to the Laplace Transformation. Inverse Laplace Transform ations The inverse Laplace Transform ation of a function F (s) is a function f (t ...

#### lecture 3 notes - time and frequency analysis

CSU Channel Islands, MAE 106
Excerpt: ... umbers. If the real part of any of the poles is greater than zero (i.e. the pole lies in the right half of the complex plane), then the system is unstable. You can prove this by taking the inverse Laplace transform of the transfer function poles in the right half plane inverse transform to exponentials in time that "blow up". Zeros: the roots of the numerator of a transfer function. Partial fraction expansion: a method for finding the inverse Laplace transform of a signal or system. The method involves expanding a transfer function with a high-order characteristic polynomial (and an unknown inverse Laplace transform ) into a sum of lower-order transfer functions (for which the inverse Laplace transform s are known). Bode plot: a common way to plot the frequency response of a system. A Bode plot has two plots. The first plot is the "magnitude response plot" that shows how the system scales a sinusoidal input as a function of its frequency. A Bode plot shows the magnitude response by plotting the scaling factor ...

#### L74

Texas A&M, MATH 308
Excerpt: ... Fall 2003 Math 308/501502 7 Laplace Transforms 7.4 Inverse Laplace Transform c 2003, Art Belmonte Mon, 13/Oct Summary Theorem If f and g are continuous functions whose Laplace transforms are such that L { f } (s) = L {g} (s) for s > a, then f (t) = g(t) for all t > 0. Definition For a continuous function f of exponential order whose Laplace transform is F, we call f the inverse Laplace transform of F and write f = L-1 {F}. (The preceding theorem is what makes this definition possible.) Linearity Property square, and algebraic manipulation before finally employing table lookup. On the other hand, the MATLAB Symbolic Math Toolbox (SMT) command ilaplace computes inverse Laplace transform s at one fell swoop. In between these two extremes is a middle ground, where one mimicks hand work semiautomatically with the new cpf (convert to partial fractions) MATLAB rountine I wrote for you along the SMT command expand. This is how some of the Hand Examples below were computed, as shown in MATLAB files in the MATLAB ...

#### ws1

Rose-Hulman, MA 213
Excerpt: ... Applied Mathematics III (MA 213), Winter Quarter, 1998-99 WorkSheet 1 1) Find the Laplace transform of ft = t by hand. 2) Find the Laplace transform of ft = ut ? 3 by hand. 3) Find the Laplace transform of the solution to the following differential equation. Note that this means that you need not find the solution at this time. Do this completely by hand and show all steps. y v ? 3y = 9, y0 = 12. 4) Using the table on page 132 in the notes, find the inverse Laplace transform s of the following ?2s functions: 4 , 23s , 2es s?5 s +5 5) Derive the formula for the Laplace transform of f vv t. (again see the table for the answer). 6) Find the inverse Laplace transform s of 1 and 2 3 . s 2 ? 2s s + 3s + 2 ...

#### SignalsHW15S07

RPI, ECSE 2410
Excerpt: ... Assignment #15 ECSE-2410 Signals & Systems - Spring 2007 Fri 03/30/07 1(28). Find the Laplace transform of 1 1< t < 3 (a)(8) x a (t) = . 0 else (b)(10) xb (t) = e-(t - 2) u(t - 3) . (c)(10) xc (t) = 5 e-2 t cos( t + 45 ) u(t) . 2(30). Find the inverse Laplace transform for the signals below. Note. You must use properties. No integration is possible. 1 - e -2 s (a) (10) Find xa (t) , the inverse Laplace transform of X a ( s ) = . 1+ s (b) (10) Find xb (t), the inverse Laplace transform of X b (s) = 2 s - 2s + 1 . s ( s2 + 4 ) s e -4 s (c) (10) Find xc (t) , the inverse Laplace transform of X c (s) = 2 . s + 5s + 6 3(15). Given the transfer function, H(s) = 10 , find the impulse response. ( s + s + 16) ( 1 + s) 2 4(15). A second-order system is described by the differential equation d 2 y (t ) dy (t ) K +2 + y (t ) = x(t ) , where K is a constant. 2 dt dt (a) (5 pts) Find the value of K that will make the system critically damped. (b) (5 pts) If K = 4, find the impulse response, h(t ) , f ...

#### finalreview

Old Dominion, J 3
Excerpt: ... cations: improper integrals, Fourier integrals, inverse Laplace transform s, integrals involving branch points/cuts* k=1 For a complete set of lecture notes, please go to: http:/www.lions.odu.edu/j3wang/math422Sp08.html 1 Review problems (Laurent series, applications or residues) 1. Find all possible Laurent series of f (z) = region of convergence for each of these series. 1 at the point z0 = 1 . Specify the 1 z2 1 at the (z i)(z 2) 2. Find all the possible Taylor/Laurent series expansions of f (z) = point z0 = 0 . 3. Show that when 0 < |z| < 1 , ez 1 5 1 = + 1 z z2 + 3 z+z z 2 6 4. Evaluate the improper integral 0 x2 dx +1 dx + 1)2 5. Evaluate the improper integral 0 (x2 6. Evaluate the Fourier integral 0 2 cos(ax) dx (a > 0) x2 + 1 d 5 + 4 sin 2s 2 (s + 1)(s2 + 2s + 5) s 2 a2 (s2 + a2 )2 (a > 0) 7. Evaluate the denite integral 0 8. Find the inverse Laplace transform of the function F (s) = 9. Find the inverse Laplace transform ...

#### EEE202_Lec13 [Compatibility Mode]

ASU, ECE 202
Excerpt: ... Lecture 13. Inverse Laplace Transform ation Inverse Laplace Transform P l Polynomials i l Roots, zeros and poles p Complex numbers Step & Delta functions 1 Solving Differential Equations g q Example: For zero initial conditions, solve d 2 y (t ) d y (t ) + 11 + 30 y (t ) = 4 u (t ) 2 dt dt Laplace transform approach automatically includes initial conditions in the solution L d y(t ) = s Y(s) y(0) dt d 2 y (t ) L 2 = s 2 Y(s) s y(0) y' (0) dt 2 Solving Differential Equations ( g q (contd) ) Laplace transform of the equation 4 s Y ( s ) sy (0) y ' (0) + 11sY ( s ) 11y (0) + 30Y ( s ) = s 2 s 2 y (0) + s ( y ' (0) + 11) + 4 Y (s) = s ( s 2 + 11s + 30) Easy to solve the differential equation in Laplace space, but needs to transform the solution to real space! p Inverse Laplace Transform p 3 Inverse Laplace Transform p Consider that F(s) is a ratio of polynomial expressions N ( s) F ( s) = D ( s) ...

#### 28

BYU, MATH 303
Excerpt: ... 28 Unit Step Function. Second Shifting Theorem. Diracs Delta Function (Section 5.3: 1 day) Outcomes: A. Dene the unit step function (or Heaviside function) and the dirac delta function. B. Recall and prove the Second Shifting theorem; t-shifting. C. Find the Laplace transforms of piecewise dened functions and functions involving the unit step function. D. Find inverse Laplace transform s involving the unit step function. E. Solve initial value problems arising from systems having discontinuous or impulse inputs. Reading: Section 5.3 Homework: 5.3: N1,N2,2,6,8,9,16,17,25,27,29,31,33 Supplementary Problems: N1. Dene (a) the unit step function (b) the dirac delta function N2. Study the second shifting theorem. Write it from memory. Outcome Mapping: A. B. C. D. E. N1 N2 2,6,8,9 16,17 22,25,27,29 Lecture: ...

#### q10

UNL, MATH 221
Excerpt: ... Math 221-05 Quiz 10 Name: Score: Instructions: You must show supporting work to receive full and partial credits. No text book, notes, formula sheets allowed. 5t e , 0t3 0, t3 1(3pts) Use definition to find the Laplace transform of the function f (t) = 2(3pts) Find the Laplace transform of the function f (t) = (2 + t)2 . 3(4pts) Find the inverse Laplace transform of the function 5 . (a) F (s) = (s + 2)3 F (s) = s2 4 . - 4s + 8 END ...

#### 27

BYU, MATH 303
Excerpt: ... 27 Transforms of Derivatives and Integrals (Sections 5.2: 1 day) Outcomes: A. Solve initial value problems using the Laplace transform. B. Determine the Laplace transform of the integral of a function given the Laplace transform of the function. C. Apply integration to determine inverse Laplace transform s. Reading: Section 5.2 Homework: 5.2: 1,3,5,6,9,12,14,19 Outcome Mapping: A. 1, 3, 5, 6, 9 B. 12, 14, 19. C. 12, 14, 19. Lecture: ...

#### 31

BYU, MATH 303
Excerpt: ... 31 Partial Fractions. Dierential Equations (Section 5.6: 1.5 day) Outcomes: A. Determine the partial fraction decomposition of a rational expression in the cases of unrepeated factors, repeated factors, unrepeated complex factors and repeated complex factors. B. Determine the inverse Laplace transform of a rational expression by partial fraction reduction. Reading: Section 5.6 Homework: 5.6: 2,3,6,9,11,13,14 Outcome Mapping: A. 2,3,6,9,14 B. 11,13,14 Lecture: ...

#### Lecture2

Pittsburgh, ENGR 1673
Excerpt: ... and exp pairs. (2) Multiplication of polynomials. (3) Compact representation of data. Difficult Solutions of initialvalue problems Easy Inverse LT Solutions of algebra problems 19 20 Laplace transform The Laplace transform of a function f(t) is defined as F ( s ) = L[ f ( t )] = f ( t )e dt 0 st Laplace transform The Laplace transform of a function f(t) is defined as F ( s ) = L[ f (t )] = f (t )e st dt 0 The inverse Laplace transform is given by f (t ) = L1[ F ( s)] = 1 + j F ( s )e st ds 2j j 21 22 Laplace transform The Laplace transform of a function f(t) is defined as F ( s ) = L[ f ( t )] = f (t )e st dt 0 The inverse Laplace transform is given by f (t ) = L1[ F ( s )] = 1 + F ( s )e st ds 2j Laplace transform table We seldom use the above equation to calculate an inverse Laplace transform ; instead we use the equation of Laplace transform to construct a table of transforms for u ...

#### Lecture2_ECE1673

Pittsburgh, ENGR 2646
Excerpt: ... lving linear differential equations with constant coefficients Example Classical method Laplace transform Initial-value problems ODE's or PDE's LT Algebra problems Difficult Solutions of initialvalue problems Easy Inverse LT Solutions of algebra prblems 19 Differential equations Linear ordinary differential equations Nonlinear differential equations Solving linear differential equations with constant coefficients Example Classical method Laplace transform Examples (about "usefulness" of mathematical transforms) (1) log and exp pairs. (2) Multiplication of polynomials. (3) Compact representation of data. 20 Laplace transform The Laplace transform of a function f(t) is defined as F ( s ) = L[ f (t )] = f (t )e - st dt 0 21 Laplace transform The Laplace transform of a function f(t) is defined as F ( s ) = L[ f (t )] = f (t )e - st dt 0 The inverse Laplace transform is given by 1 + j f (t ) = L [ F ( s )] = F ( s )e st ds 2j - j -1 22 Laplace transform The Lapl ...

#### s_quiz_8

Texas A&M, MATH 308
Excerpt: ... MATH 308 Quiz 8 Name: (10 ponts) Find the inverse Laplace transform of F (s) = s-2 s 2- 2s-3 ' ...

#### t1revsp03m

Virginia Tech, MATH 4564
Excerpt: ... MATH4564 - Review Test No1 - Spring 03 - Konat e Notice: Show your work. A right answer with a bad reasoning will be considered as wrong. No calculator, no notes allowed. Abstracts allowed. - 1 Find the Laplace Transform of the function f (t) = (t + 2)2 - 3(t - 1)2 2 Find the Laplace Transform of the function f (t) = t2 e2t + e(1/2)t sin(3t) 3 Find the Laplace Transform of the function f (t) = 2e2t u5 (t) + 3e(1/2)t t-3 where u5 (t) is the step function at the point t = 5 and t-3 is the Dirac function at the point t = 3. 4 Find the Inverse Laplace Transform of 2 F (s) = (s - 3)5 5.a Calculate the derivative of 3 s + 4s + 13 5.b Derive from above the Inverse laplace transform of s+2 G(s) = 2 3 (s + 2)2 + 9 F (s) = 2 6 Solve y + 2y + 2y = h(t) y(0) = 0, y (0) = 1 and t 2 where h(t) = 0 1 for 0 t < for t < - 1 ...

#### exam2

Maryland, MATH 341
Excerpt: ... MATH 341 EXAM # 2 Instructions. Show all your work. Be sure your name is on the booklet and that you have signed the honor pledge. You may not use calculators, notes, or any other form of assistance on this exam. (1) (10 pts) Find the inverse Laplace transform of: s2 s - 4s - 12 (2) (20 pts) Use Laplace transforms to solve the initial value problem y +y = cos t 0 t /2 0 /2 t y(0) = 3 , y (0) = -1 You may leave your answer in the form of an inverse Laplace transform . (3) (20 pts) Write down recursive relations for the coefficients of a series solution to y - 2ty + y = 0 (4) (15 pts) Show that 1 are roots of the indicial equation for t2 y + ty - (1 + t)y = 0 (5) (15 pts) Express the following initial value problem as a first order system of differential equations: y + 2y - y + y = et y(0) = y (0) = 1 , y (0) = 3 Include the initial value for the system. (6) (20 pts) 1 0 . 2 1 (b) Find the solution to the initial value problem (a) Compute exp(A) for the matrix A = 1 0 x(t) 2 1 x (t) = x(0) = 2 1 Date: ...

#### Lec18_228_09 App B

Washington, PHYS 2278
Excerpt: ... Lecture 18 Appendix B (Laplace transforms) Ex 8.9:12 Here we solve an inhomogeneous differential equation (plus initial conditions) in a variety of ways, including Laplace. First proceed directly with DSolve In[1]:= Out[1]= DSolvey 't yt Exp t ...

#### Lecture2

Pittsburgh, ENGR 2646
Excerpt: ... transform ^ f ( s ) = L[ f (t )] = f (t )e - st dt 0 ^ f (t ) = L-1[ f ( s )] = 1 + j ^ f ( s )e st ds 2j - j Initial-value problems ODE's or PDE's LT Algebra problems Difficult Solutions of initialvalue problems Easy Inverse LT Solutions of algebra problems 10 Review of last lecture Important concepts from classical control Control system Feedback Dynamical system Transfer function Definition Laplace transform and inverse Laplace transform ^ f ( s ) = L[ f (t )] = f (t )e - st dt 0 ^ f (t ) = L-1[ f ( s )] = 1 + j ^ f ( s )e st ds 2j - j We seldom use the equation on the right to calculate an inverse Laplace transform ; instead we use the equation of Laplace transform to construct a table of transforms for useful time functions. Then we use the table to find the inverse transform. 11 Review of last lecture Important concepts from classical control Control system Feedback Dynamical system Transfer function Definition Laplace transform and inverse Laplace tran ...

#### PF_Handout_Part1

Midwestern State University, EE 321
Excerpt: ... EE321 Handout-Part 1 Partial fraction expansions are used to find f (t ) = { F (s)} Consider that after "Laplace transforming" your circuit equation, you have the following s-domain relationship: F (s) = N ( s ) a n s n + a n -1 s n -1 + . + a1 s + a 0 = D( s ) bm s m + bm -1 s m -1 + . + b1 s + b0 All a's and b's are real for circuit analysis. F(s) is a rational function if n and m are positive integers. F(s) is a proper rational function if m > n. Only proper rational functions can be expanded as a sum of partial fractions. 5. Long division can be applied to yield a proper rational function. Example: 1. 2. 3. 4. 9 2 - 2s -1 = ( s + 4) 2 s - 1 ( s + 4 ) s+4 2s + 8 -9 2s - 1 9 = 2- s+4 ( s + 4) 6. General approach to finding f (t ) = { F (s)} when F(s) is a proper rational function: a) b) c) d) e) Find roots of D(s) Use the results from a) to factor D(s) Set up partial fraction expansion (see examples below) Find unknown constants in partial fraction expansion Find inverse Laplace transform Cons ...

#### lecture19

McGill, MA 261
Excerpt: ... 4 Formula (III) The next formula shows how to compute the Laplace transform of f (t) in terms of the Laplace transform of f (t). L{f (t)}(s) = sL{f (t)}(s) f (0). This follows from 0 0 L{f (t)}(s) = 0 est f (t)dt = est f (t)| + s 0 est f (t)dt = s est f (t)dt f (0) since est f (t) converges to 0 as t + in the domain of denition of the Laplace transform of f (t). To ensure that the rst integral is dened, we have to assume f (t) is piecewise continuous. Repeated applications of this formula give L{f (n) (t)}(s) = sn L{f (t)}(s) sn1 f (0) sn2 f (0) f n1 (0). The following theorem is important for the application of the Laplace transform to dierential equations. 3 3.1 Inverse Laplace Transform Theorem: L{f (t)} = L{g(t)} = f = g. If f (t), g(t) are normalized piecewise continuous functions of exponential order then 3.2 Denition If F (s) is the Laplace of the normalized piecewise continuous function f (t) ...