Question

III. (10) When solving this problem show all the steps needed to transform the expressions into ones that can be found in the table and indicate the entry of the table used in each step. a) Find the Laplace transform F(s) of the function (3 - e2+ 5e-6) sin(74) b) Find the inverse Laplace transform f(t) of the function F(S) = 9 $2 + 8 - 20

Answer 1

boal (3- ett geot) sin(at) SOLS = 1 { (3-624+se6t) sencat)} Expond, L { 3 SPM (77) et senzt) + 5ē soch Spacht)} =3L{ Bsin (7t)}L atsec 743} +5Lserole sin(7} find, L{ spoczto} use Laplale transform table: l { sencats} staz a 7 - s²+49 L{2SPAC703 Pf L {fct)} = FCs) then Apply transform rule: Lę eat et fct)} FCS-a) for et sinc7t); f(t)= senc7t), a=2 Lęsincat)} 7 - s2+49 ㅋ (S-2)2 +49 알 Spoczt)} and, LE (5+61 +49

from 0, 7 7 = 3 t 5 s?+49 (5-2)+49 (5+6+49 Refine 21 ㅋ :: FCS) = 35 t s2 +49 (5-2²+49 (S+6)?+49 b sols FCs) ។ Sa+S-20 ។ S²+5-20 9 Take the partial fraction of S2+5-20 9 t В S+5 (5-4)(5+4) - ACS+5) + BCS-4) for the denominator root 4: 9 A = 1 for the denominato root - 5 Boa) (-1) st5 stat = I' { $tum - S5} = I' { šy} I'&sts}

use Piuorse Laplace transform table I'{}=e 4t è st = e at -st fct) e Thank you.