Question

2. (4) For each question, use Th. 7.1.1 to compute (0) Show your work. Write your answer in the box (a) f(t) = cos 21 L{f(t)} = (b) (t) = +10 L {f} (c) / (t) = 5e -3r 44 sinh 34 L{f} (d)/(t) = 6+*+ 7 sin 40 LS (0) =

Answer 1

Leming 1-1) [ { cosat} s S2492 sinat S²ta 2 e-st e-iat de Inef- Ideriaty g e-(s+a) at 8 e-tp dt i p=stia o to e-tp 7 -P Jo Р stia Sola s²ta? ia s S' +2 sete2 I sia 1 leiate = s sta sta 527a2 s²te2 S S²492 i S2492 1{ cosat - ismat Compare real and imaginary part , we gu. lcosatz & lesinat} S S2to2 a $2+2

Put value of ② 4 6 in 4f1t)y 51 *{f(t)} = 5 S+3 +43 S²- 12 S29 S+ 3 () flt)= 6+3+ 7 Sin 4t. 1 {f(t)} = 1 {6t3 +7 sinut y = 6 1 2 + 3}+7 L{ sinuty 1+3} = Jest e-st t² dt fake st=u Sat=dre 2 3 da fem u jo da S. 3+1 da=1 13ti 4 11 - Jest 84 $4 s' (A 11t3y = 31 SY 4 à s²ta? 4 S²+16 B L { sinat? (By Lemma) Il sinuty @ 4® in L {f(t) = 6. 31 Rafflt)) = 34 Put value of + 7.4 S2+16 S4 S។ 82716

L { flt)y= (0) L {5e3t + 4 sinh3t } - 5 Lie 3t Z + 4 hy sinh 3 t} Le-3t} = { este = fe (stat att -(5+3) + (5+3) = O + 1 I St3 if S>3 I S43 (A L {e-3tl, I 8+3 And I sin h3t} = 1 / 1 (ezt - e-3t)} - Į tenty-1 a pesty - ferst ezt dt - f e-ste 37 dt 2 8 -(3-3) dt - - (St)t dt e 9 e a ㅗ 2. e-is-37 e-(S13) - (5-3) le 75+3) g 2 St3 į S-3 $+3-83 S²-9 3 S²9 e sim hat} = 3 B 8²

Solution - a f(t) = cosat L{cosat = s Sata? (By Lemma) Put a=2 L {cosatz S S2+4 co bfit) ta +0 L {f(t)} = { to to} = Loft2} + L/10} ht2y + lo hi By linearity property equifilt) +6, falt)y= GI{ filt)y + c hi falt)} Bet steg Jeans an de las te het sut=da. o 61 1 e-9344-1 da = T3 2 53 53 S3 0 I to y a 2 CA s3 Now, ho}= 10L{1} = 10 jest. ldt = get 1210} B to ut value of ( 4 (B) in li If th Ho} = a + b