Question

Use the Laplace transform table and the linearity of the Laplace transform to determine the following transform. Complete parts a and b below. {{4tº e -88 – est cos 3t} Click the icon to view the Laplace transform table. a. Determine the formula for the Laplace transform. 244te -8t - 25+cos 13t} = (Type an expression using s as the variable.) b. What is the restriction on s? (Type an integer or a fraction.) S>

Answer 1

We know LEP (2)} = jest f(t) at, where syo. if flt) = An eat Then > 12 A(H)} = joo st th eat dt S O and (S-a)t dt P (n) (say) Taking ecs-a)t as a & to as 1st function using integration by parts, we get (s-alt en é (s-alt d+ - (s-a) -(s.a) -(s-alt 4 - (5.a) d e an é - Cs.at : 0-0- J m. tra è (5.a)t - (s.a) n-1 st (S-a)t dA e S-a lium th elsaalt -(sna) n P (n-1) ma 1. Here m L {f(t)} = P(m) P(-1) repeating this we get Panda niin-1 (snage p(n-2) sca

P(n) = n. (n-1). (M-2). (s-an P(1) PIO) (sagn (s-alt Now, Plo) do di 0 e-s.at -(S-a) a(5.a) sra lim elsne 0 -Gra) Then P(n) = ก ) m) . (1). n+1 {th cat] = S-a) if flt) = eat cos bit L L f (l)} Now Then . est cusht dt 0 e (s-alt cosbt dt I taking & Cs.at as and function & cosbt as first function, using integration by Parts we get

cosbt. e Cs.at (Cosbt) -(s-a)t dt - (5.a) (s-a) 0 o- (s-a)t -(s-a) b simbt dt -(S-a) 6 S-a (S-a) t s-a simbté O est & Again taking simbt as function, using integration by parts we get] e-S-a) f as and (6.at 6 e Sinbt dt s-a - (5.a) b cus bt: -(S-a) (S-a)t 6 (0-0) + cosete e s-a S-a - L {f(d)} s-a s-ar b {f(t)} L2 f (2)} s-ajr 2 is as to > {flt)} = > L { $(+} fahrt 62 at cosht} = S-a () LI (s-a) for

Then L {4x² -87 _ est cos 3 t} 4 L { t - St} L { est cos Bt} ELS is linear S-5 3+1 - 4. 3 (S-(-8)) (5-5) + (3)" [using ) & (2) (n=3, a 8) (a=5, b=53) 24 (5.5) (5.5) 3 (5+8)4 Here) s y