Question

Find the inverse Laplace transform

a . 3 s4 - 2s s2 (3s2 + 4) 3 S4 – 25 (s + 1)(3s2 + 4s + 2) b.

Answer 1

S (33 - 2) 3st – 25 S21352 +4) S2 (32+4) ir not les Since the degree of the denominator than degree of the denominator of persom polynomial long divison'a ght3332) S(353-2) -482-25 It 32(352+4) 83 (35+4) G Now we do partial traction of- - 45²-25 - 45²-25 3 (35+4) 382 (3+ 4/3) 45+2 353 +45 ductor the clonomastector - -45-2 45-2 33+45 the Bstc 35 (547) form of partial fraction decomposition is -45- s (8 + 4 / 4 multiply both side by S(2+4/3) "A (5+4 /3) + (85+ C) 5 aly + 8+ 4 3 45

45 - 2 3 2. 3 3 s (BS+C)+ (35244 ) A 2 Ass 52 ( 3A+387 +350 +48 3 euating the coefficient 3A+ 3B=0 and A=-B 36=-2/2 and 4A= -2 9 2 and TB = 7 There for,

- 4S- let F(5)= 3 s(52+ 4/5 35²44 7 3876 - 5 / 5 s 4 3 Ś Š + 1/2 32 44/12 FCS) 52462 2/35 52423 1-4*++ inga sistema both side - 253 4.3 3+ (2 7 9 How fake laplace inverse ['(FC) - 41'1t) + Closed f(t) = 1:1 1 + 7. cos ( 2 / 34 23 sin ( 2 ) f(t) = + 1 + 1 casi (as ( 2 + 3) - 233 sin ( 2 4 / Now from 354-7 cos (es) 20 in S² (35+4) {cas (Zz ! ) - 203. sin ( 2 (Igst 1 / 25 + FC? 6 % 5 6 لها. F(t.) 6 ..t Sta2 cos cab). sincat) .. t' (star