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  1. services.math.duke.edu › ~jtwong › math353-2018MATH 353 LECTURE NOTES LAPLACE TRANSFORM: FUNDAMENTALS - Duke...

    Let Y (s) = L[y(t)] be the Laplace transform of the solution. Applying L to the equation, we obtain the transformed equation L[0] = L[y0] L [y] = sY y(0) Y: Since L[0] = 0; we get. 0 = (s 1)Y y(0); which is trivial to solve! The transformed solution to the ODE is then y(0) Y (s) = :

  2. www.integral-table.com › downloads › LaplaceTableTable of Laplace Transforms - Integral Table

    Table of Laplace Transforms f(t) L[f(t)] = F(s) 1 1 s (1) eatf(t) F(s a) (2) U(t a) e as s (3) f(t a)U(t a) e asF(s) (4) (t) 1 (5) (t stt 0) e 0 (6) tnf(t) ( 1)n dnF(s) dsn (7) ... « 2011 B.E.Shapiro forintegral-table.com. This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Revised with ...

  3. math.stanford.edu › ~ralph › math53Laplace transforms table - Stanford University

    Laplace transforms table Function Laplace transform eat 1 s−a tn n! sn+1 sin(at) a s 2+a cos(at) s s2 +a2 δ 0(t) 1 y0 sY(s)−y(0) y00 s2Y(s)−sy(0)−y0(0) eatf(t) F(s−a) tnf(t) (−1)nF(n)(s) H(t−c)f(t−c) e−csF(s) (f ∗g)(t) F(s)·G(s) ecttn n! (s−c)n+1 ect sin(at) a (s−c)2 +a2 ect cos(at) s−c

  4. www.math.purdue.edu › ~caiz › MA527-caiHome - Department of Mathematics - Purdue University

    State the Laplace transforms of a few simple functions from memory. What are the steps of solving an ODE by the Laplace transform? In what cases of solving ODEs is the present method preferable to that in Chap. 2? What property of the Laplace transform is crucial in solving ODEs? = Explain. When and how do you use the unit step function and

  5. ocw.mit.edu › a27d2c4760ffdb580bc80828367a9865_laplacetransformTable 1: Table of Laplace Transforms - MIT OpenCourseWare

    Table 1: Properties of Laplace Transforms Number Time Function Laplace Transform Property 1 αf1(t)+βf2(t) αF1(s)+βF2(s) Superposition 2 f(t − T)us(t − T) F (s)e − sT; T ≥ 0 Time delay 3 f(at) 1 a F ( s a); a>0 Time scaling 4 e − atf(t) F (s + a) Shift in frequency 5 6 df (t) dt d2f(t) dt2 sF −(s) − f(0 ) s 2 F (s) − −sf(0−) − f (1)(0 ) First-order differentiation

  6. sites.math.washington.edu › ~marshall › math_135Tableof Laplace Transforms - University of Washington

    Table of Laplace Transforms. In the table below c is a constant. The functions f and g are piecewise continuous functions of exponential type; F and G denote their Laplace transforms respectively. The Heavyside function u0(t) is defined to be equal to 1 for t > 0 and equal to 0 for t < 0, and δ0 denotes the δ-“function” at 0.

  7. nreadin.math.ncsu.edu › 341 › tableTable of Laplace transforms - North Carolina State University

    Table of Laplace transforms f(t) L(f(t)) or F(s) 1. 1 1 s 2. eat 1 s−a 3. tn n! sn+1 n≥0 integer 4. eattn n! (s−a)n+1 n≥0 integer 5. sinkt k s2 +k2 6. coskt s s2 +k2 7. eatsinkt k (s−a)2 +k2 8. eatcoskt s−a (s−a)2 +k2 9. 1 √ t r π s 10. u(t−a) e−as s a≥0 11. δ(t−a) e−as a≥0