Question

1. Basic properties of Laplace transforms: Show all of your steps. You can use tables of Laplace transforms to assist with the calculations. (a) Using a table of Laplace transforms, evaluate L {2t3 - 3e-2 t (b) Evaluate the Laplace transform of t2 sin(bt). d2 ds2 Hint: First verify the identity estt2 f(t)dt = estf(t)dt

Answer 1

al 1(a) L{2 t - 3e ?t qua = 9 24 + ²) - 3.250 2t} {. 2 is linean) T - 3X1 :21#) -n! = 2 * 3! st +2 l leat) - S L -a 32*6 - 3 12 - 3 I (6) Verify the idendity est f(t) do d² st f(t) all dsds RHS Let Flo= 10e-t fit) da =AFB) = f(est dt = f fit) d le 31 dt - 1.3 - 3®ft f(A)] east de F(S) = -4° [f()] est det Hotel FB) = a b [447] 244 ) - 3-40 [4 (8) - Go Oficts est akn = . Hence d² F(S) = d² post f(f) dt = tf(t) est at djz a fech) dt noe

Hansformation we need to find Laplace How (رط )وزی 2 و ع (h sink) له (و لا = (طع ش د ورود =--------- بط= (sEH= لاغا که ا) ] 4 - as last 8²72 | - 2 ) - (0) (+) ) م و - ] به رد x رد Xx - ( 3 ) ا - = (83) 4 آ رہا - شاهد ] ( ) 3-= * ره) م3 ماوی + + --- ای ) - = ( و - = L [ gis ,