Question

B. Find the Laplace transform of the following functions. 1. f(t) = {" where n is a positive integer, 2. f(t) = { • 10<t< 0 <t< 3. f(t) = { t 0<t<1 2-t i<t<2 0 2<t<

Answer 1

1] where is a positive integer. Let f(t) = tn Then litz L { f(t)} = n! shti 2] f(t) = Let oct<T T et cool i fit) = 1 (UCE-0)- U(t-1)) to cult-ID)) Ult) - U (HE-I) apply laplace transfo am on both sides - L{f(t)} = Ist-o} - lu { t - 11 ) } Fcs) f(t) = - TS - Léos s ک FCS) = Ś [t - éts -TS [i-e

t ostal Let f(t) = 2- t 1st & 2 t72 The was function is written as a unit step function fia = t (uct-o) -U(t-1)) +(2-0) Cuttt-1) - U(t-2)). to (uct-2)) f(t) t ulto - turt-1) + Zulte 1) -t (u-1) -zult-2) + E luct-2)) f(t) = t ult) -2(t-1) ult-1) + Ct-2) Ult-2) apply laplace tansform on both sides, we have -25 -os e -2 és + 5² = F(S) = [1 - 2es + e 2]