Question

Find the Laplace transform of each of the following functions. 1. $(t) = f*(4(t – 1)* sin(67) dt L{v(t)}(s) = b. g(t) = [ e 2-3(t-1) cos(71) dT L{$(t)}(s) = c. y(t) = e5(t-1) sin(97) cos(6(t – T)) dt L{s(t)}(s) =

Answer 1

1968=S(4 C+-2))"sin (cz) dt Let f (t) = (4+1² g(t) = sin 67. Then 46) s f(+2) gce) de f&g (4 (+) [ convolution] Then L(&(t)) L(f) L(?) 2(1) L (16+2) 16 L(+2) 11 22 16 ["L(+"). $?? 2+1 Sa . 32 s L(7) = ( sine 6+) L(sinat) a s²t az S+ 36 J 46°) - LCF) L(8) 32 12 2 6 s2+36 s” (s436)

b 46477 23t-t) -3 (t-t) cos (77) de Let -37 е. fet) = get) cost Then $(t-e) geelde f = g (4) [ convolutions Then L(&(4) L (f) L(8) L(f) = L(63t) 5 +3 L68)= L ( cos 7+) S (sat) s²+ 49 set az ] & LC4) = L(f) L(9) S s²+49 (5+3) S (5+3) (3²449)

c) 466) = *250-elsinge) epslect_ty} de 50t-c) - Le cos (6(1-2)) sin (91) de 0 let f(t) = SE cos (6+) 96) sin (9+) Then t VH) 7= $(6-c) zce) de st fog C+] [ conuciulion] - LC9) LCH) L(3), LA - COS (6+) s-5 Lleatf(t)) (5-5)2+36 F (S-a) whene F($) - LCH) s²_10s +61 L(cosat) L(8) = L (singt) S-5 and S - s²+ a2 ราย S2+ 31 : L(sinat) a s²taz - LC4) = L(f) L(g) s-5 SP-105 +61 s2+81 9(5-5) (s +81)(s<105+6)