Question

Use the method of Laplace transforms and the accompanying proof results to solve the initial value problem. + y'' +9y' + 18y = f(t); y(O) = 0, y'(0) = 0 Here, f(t) is the periodic function defined in the graph to the right. Click the icon to review the results of a proof. Choose the correct answer to the initial value problem below. 1 O A. y(t) == (1-2-3t) 2 6 3 B. y(t) = - Ite - 5t t(t + 3)(t+6) 0 1 OC. y(t) == E (1-e - 3(t – 10n)) uct – 10n) n = 0 00 1 OD. y(t) = 3 (-1)^(1-e –311 – 5) 2u(t – 5n) n=0

Answer 1

noy if f(t+1) = f(t); fet) is function nok with Pri od 7 L ( f(t) = f (t) est de ng T 27 st - fltl et du I f(t) est dt + f f(t) et dt T ( not ♡ do J fet) est ntzt n=1 (n-17 let t = kt(n-1) T =) df = dk 2 fac f (K + (n-1)T) = f(K) t=nT =) k=.T. t= (0-1 T =) K=0, T -SK - (n-iT dk e so, 2(f(t)) i s f(K+ (0-1)t).e D

hlf(t)) = 8W -(0-1BT e Jeck). est de =) 2 n-- (n-1ST stores j f(t) est dt. E ne) 7 S feljest do -54 I Given D.E y" + qy't sy = f(t); y Lo=0, y'Coleo lef Lly(t) = y(s) Lly'(t)) = Sycs) – y los Syu) Icy"C)) S(SYS)-410)) - Yo) Ŝyes) c. fit) thoven 3 > t 10 5 3 ost Ls With Period f(t) = 5 5e talo T=80

L (fit) a So, I fit), est at 1 e los Sa estan (570) (1-é5) cité55) E = gust X S (l-e -SS ) lipess) 1 3(less) X (bess) (itess) E s(1+e5s ) T Now [(")+9LCY") + 10 Lry)= a (fet)) ( 54+95+18)yes) : 3 slitess) 3{ (-1) sasn notes I tess no -Sns Yes) = ع (۱) کے 능 s(8²+95 +18) n-o ~ +93 + 18 3² +65 +35716 = S(+6) +3(316) (5+3)(3+6) 2

let K A B St3 Sad sis+15+18) S A ( ² +95 +18) +B(2+651+ c(²+3) s(32+95 + 18) (A+B+C) } + GA +6B +3C) + 18 A s(+95 +18) =) A = ter By comparing: 18A = at 68 +36= -1 9A +GB+36=0 A+B tcão 76/BtC a ($res -!) 18 + -32= -1 2 + 3 rie =) (= 1 a) B - 18 9 ㅗ 55 +95 +18) 185 1865+6) 9(5+3) 2 st3 + is [s sti] 18

So, Y(c) = 3. [ Ž + ج ۱۱-) 2 S+3 stro) 2 a) YCS) = 1 & Chests (3 sh + )] st3 sto L (uct-c). fct-c)) els L (fit)) 2 tl lets. 2 (f(t)) = uite).fct-c) d' Y(S)) = 1 [ { L. ( 13 -2 143 576 Here 2(f(t)) = - V + st3 5t6 V V -64 tê 2) fct) - 2 6 37 (1-23+)? So y(t) = 1 [ EL)". ult-5n). f(+-50)] no lyct) = 1 il vºuch-sn). (1-284-sa)" ] option D