Question

Find the time function corresponding to each of the following Laplace domain functions. Use the proper partial fraction expansion (PFE) when necessary then use the Laplace tables. 10 la 8(8 + 1)(8 + 10) 2s + 4 (b) F() = (8 + 1)(2+4) (C) 53 +352 +58 +8 (8) = (x + 1)(2 +9)(2+28 + 10) - Doozy.

Answer 1

(a) F(S) = 10 s(s+i)(8 +10) Now, let 10 SIS+)/S+0) A + B Sti c sto - - - 10 = Als+i)(8 +10) +B(s) (5 +10) +(s)(5+1) of Sa 103 10- A (0+1) (0+10) + B.(0) 10+10) + C (ollo+1) 10= 10A = At S=1 lo= A.O +B(-1) (-1+ 10) fc. o => 10--98 -> Bath At S=-10 10 = A.0 +B.0 + c(-10) (-10 +1) 10 = 900 =)

Therefore 10 sistis to e te voeg togalerie) - Fls) - /3 - 1 st) + atstio) Apply inverse Leaplane transform bush sides, we get stoly I Fell = La Plody = "hty - 10 Ly ty + - Flla 1 - 10 et ta e lot (6) F= 2 + 4 25+ 4 Els = Let Let 25+4 = 25+4 (ts) (5²+4) - A + BS te + BS+C 82 +4 St1 28+4 = A(8°+4) + (85+c) (s +1) >> =) 28 +4 = Als?+4) +B(si+s) + C($+1)

ci comparing coefficient of power of 's' both sidel, we get sa O= A B = A=-B -- S a B ---Cii) Constant) 4 = 4A te --- (11) term . By ci) and (ii) , we get 2= -Atc --- (iv) Now, apply operation 4 (lv) + (iki), we get 50=12 By Civi, 2 = A + 12 =) - 1 2 - 2 = 12 - 10 Therefore 25+ 4 + -25+12 5188 (St) (5 +4) 5(5+1)

FISI= - + 18-as 5(5+) 5(+4) FRESCHI - Bion) Bbbly prvense Laplace toonstrom both sides, we ye FG1= L4 PYTLy stry to 2 144 544) =) pela + sinat a coiat - PA - B et t Sinat. Cosat