Laplace-Table - fimficew “EV/2766’ Mdflémaf/‘kr...

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Unformatted text preview: fimficew “EV/2766’??? Mdflémaf/‘kr, f-A’myszf'f, oofézzfd _ 296 ’ Laplace Transforms 50,8 Laplace Transformzfimerai Formulas an m) = $30)} = f ammo? Definition of Transform 0 fl!) = 3'1{F(S)} ' Inverse Transform Elam) +~ bgm} = aEBUtO} + wow} ${e°"f(r)} = F0 — a) .r—Shifting 3-1%“ _a)} m emf“) {First Shifting Theorem) $0“) = sin) m 1(0) 580°”) = 52336) — Sflo) ”' f’m) Differentiation 30mm) m 37:58”) _ sn~1f(0) _, . , . of Function "' — f(”—1)(0} t g f0”) €17} m ~:j‘ii’U) Integration of Function 9 ${f(r — a)u(t »~ a)} = e'“F(s) t—Shifting gfllwwflmsn 2 f0, _ (0W m a) (Second Shifting Theorem) Ettf (1‘)} = MF'(s) Differentiation of Transform 3 = I FG) d? Integration of Transform ’ s 3 (f * gm) = from: — «rm 0 t r m I fit _ T)g(1,)df Convoiutron 0 $0” * g) = $U)§B(g) f Periorfic with Period p Table of Laplace Transforms .5 t“'”1/{n ~ 1)! n 1/ V m 2% mm) 3 lction m" sin cut a) heorem) cos wt 1 . ‘ransform - smh at msform m E; {I "- cos m!) "'L( t ' r) as to 5111 w . 1 . Pcnod p ——3— (sm cu: w cut cos wt} 2a) (continued) 298 V " LapEace‘Transforms "W sin wt + wrcas mt 2m{ ) l bzmaz (cos at ~ cos b1) 1 W (sin kt cos k: —- cos kt sinh kt) 1 an?" sin kt sinh kt 1 a? (sinh k! - sin kt) 1 ~27? (cash kt m (:65 kt) 1001‘} Van; 3“”(1 + 2m) v; I k—Elz m "*2: Ila—11201?) Jam/IE) 1 V'th 1 m sinh 2w; cos ZVE k 4:13:49 Win: - 'y (y ‘#= 0.5772) 1 T (ebt m eat} (continued) Chapter Review 299 2 T (1 We cos wt) 2 T {1 - cosh at) bl) %- sin wt 1. . m are cots 81(3) IS kt sinh kt) s ) Compared to the usual method, what are the advantages of the Laplace transform in solving differential equations? ‘ ‘ , 2.. What is the crucial property of the Laplace transform that makes it suitable for solving differential equations? 3. What do we mean by saying that the Laplace transfprm is a linear operation? Why is this I practically important? 4. For what problems would you prefer the Laplace transform over the usual method? Give a reason. 5. What is the subsidiary equation? How is it used? 6. Does every continuous function have a Laplace transform? Give a reason or a counterexample. 7. What is the unit step function? Why is it important? 8. What is Dirac's delta function? How did we use it? 412(0‘} 9. State the Laplace transforms of a few simple functions from memory. ‘ 10. State the formula for the Laplace transform of the nth derivative of a functiOn fit) from memory. 11. Can a discontinuous function have a Laplace transform? (Give a reason for your answer.) 12. Does tan 1' have a Laplace transform? Is it piecewise continuous? 13. If you know fir) m 2'1[F(s)}, how would you find EE"I[F(3)IS2}? 14. Is §£{f(r)g(r)} W ${f(t)}$[g{r)}‘? Or what? 15. What is the difference in the shifting by the first shifting theorem and by the second shifting theorem? Laplace Transform. Find the Laplace transform of the given. function. (Show the details of your work.) 16. e"sio art 17. c0532: 18. Sin2(7rI/2) 19. 3‘14: — 2) 20. t2u(t ~ a) 21. t* e'3‘ 22. a” at cos 4; 23. cosh 1%: 24. rcosr + sin! Inverse Laplace Transform. In Probs. 25m33 find the inverse Laplace transform of the given .5772) function. (Show your work.) 3+3 1 s+1 25..—— 2. .. -s2+9 fiszmhufi s ...
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