Question

Find the inverse Laplace transform, £^{F(s)}, of each of the following functions. Be sure to show all your worl 1. F(S) (s+2)3 = s+1 2. F(s) = $2 +48 $2 3. F(s) = (8-1)(8+1)(s+2) • 4. F(s) 6s+3 $4+592 +4 5. F(s) = 16 1 S 6. F(s) (s+2)(s2+4)

Answer 1

Given F(S) (s +2)3 4 로 (a+bB_ 9343a2bf39b4b3 FCS) s + 3s2(2) + 3s (2) 4 8 t s+ 8 2 + F(S) S 6 + 12 S2 S 3 9 (F(3) + S s2 Laplace transform + 3. 들 + + 7 linearity property of inverse llar) + F③J a디K(SJ + B년 월(3) t() + () + 1 네 이 12. 9 (F(s)) s3 1 8 + 월 54 s3 '( ) + 6 IT() + 2 t 8U + s4 머 - t . J (-1)! 2 t 12 +8 +3 46t + 3 2!. LF(s)) g] 2 + 오f3 1t6t +6t 3 L I s +) = 1 (s-1) ㅔ s(s+4) F(S) ㄴ (2) - s2ts st4 3 3s44) L

3. 너 15+4) s(S+4) S (s +4) [글 - ] S44 이 ㅔ 1 4 s 3 수 S s + 4) (wing linarity propenry 4-3 3 10 - + 늘 (t 을 2 S+) 너 (+4) [ ( ++ 1) + (1) 4 t 44 3e98 s-a at 대 SH s244s ㅗt 수. + 3 t'(F(4)) = tilte) (+2) (+1) wsing partial partial Fraction 2. S t의 ㄷ A (S1) + B (+) (+2) (S-1) ( s +2) (S+ 1) s2 A (SI)(+2) + (S-1) Cs+2) + c(s-1)(s+1) put s=1 in equation o A (2) (3) +BO 4 Co 12 = A put s - 1 in equation o Ao + B(-2)(1) + C. 0 네2 - 11 IN LI

put 2 in equation ④ + 80 + C (-3) ( -1) 2 (-2) . 쓸 C 3 (51) (s + I) (s +2) LI (3-1) (s +) 6+2) . + 43 3 # 中 (Sty + -2라 ㅎ e 놓으라 २ t -2士 e + 4 3 2. (4) t'lF(S)) 너 6543 (st + 532 +4) (CS+3) s4t 452 ts244 ( S +3) 3213244+1(5244) LI 6S+3 (s2+1) (s² ++) 네 3. 6S 너 2+1) (²4) (41) (34. . = 6ls 3 [나] + [ ] - ) 3 -

+ cos at (F(S)] = 2 1 2 Lls 312 20s s2412 s222 네 너 s242 52422 너 S sta2 t a sinat s2492 0 -1 (s+3) = 2 C Cost- 2 (B(2t) ( 44 + 552 +4) + sint 2. 2 [ Cost Co82tJt 8int- Sin2k 2 -) 5 Lt F(S) s4-16 2 1 -6 네 (3) 2-(4) 2. = (a+b) (-) L (s2 +4) (s-4) ~ = s24 ( S422 8 1 32-2 리 sinat L- 5242 sin2t 니 Els) 11- 8 8 X 2 (5+2) 3-2) stz sinat 16 - ~ a L-1) 4LS-2 8 sinat 16 읊 LI (5+2) 32 32

I'[st 16 ezt e-28 sinat 32 32 sinzt) ce et-e-27-2 32 16 ["F(S) 2 C L [ (5+2) (3274) L-!|(5+2) (32+4) #11s+2) (3²+4) _(5+2) – (52+4) A1152 +43 (5+2) (52+4) =li s²+22 S + 2 (s't2 = [2222 +2 1448*+2 L! J (5+2) z 2 # [cos (2 t) + 2 sin(t) – е-2tJ [ coset) + sin(2E) -6-24 e-24 = Al-