Question

Compute Laplace transforms of the following functions: (a) f1 = (1 + t) (b) f2 = eat sin(bt) 11, 0<t<1, (c) f3 = -1 1<t<2, | 2, t>2, Find the functions from their Laplace transforms: (a) Lyı] s(s + 1) (s +3) 2+s (b) L[42] = 52 + 2 s +5 (c) L[y3] = Solve the following initial value problems using the Laplace transform. Confirm each solution with a Matlab plot showing the function on the interval 0 <t<5. (a) X+3y=t, y(0) = 1. (a) dy y dy = 0, where f is the unit step function: 0<t<1, otherwise." dt2 T2+ = f, (c) 479 + 4y = sin(2t), y(0) = 3, 040)=-1.

Answer 1

1) a) fi = (1++)? = itt tot => L { fiq=t{1+t²t at} = l{1}+{{+24 +2+{t} w.K.T, L{"} = nenti 10 b) f2 = cat sin(bt) => L{fs} = L { eat son 6t3} By the First shifting the onem, we have theat f(t) 4 = F(S-a) = L{sinibttsys-a = (*)s-ys0 dx - 7 S- a c) By the def. of L.T, we have L{Ft) 3 = stist fleidt 'ist{fo2 = retajet e çësitas en totale

) CENTRO CENTRO we have et=0 s tem +2(075) =(7 t/t725_272225) = $(1-zë +372), rol - e tae 2 ( 3 - 2 e + a) a) L{ Yyy = S (5+1) (5+3) Y +7 8 = ['{6+1 67) consider, sist1/573) - § +3717 543 B + = 5+1 Considen TX 3 put s=o=7 A = put s=-1 =7 BE- put $:+3=7c = M(-345 = 27 * = 1! Şov - - į t'{}}- ¢ L'{ iftara mante = () - été est

b) L{%2} = 245 s+25+5 =7 = [ -15 . - 2 stol 3725 +5 | ?? (5+1'74 z et l'{ St 1 z ét (c'{ #+{ stry] = et (coat + Ź sinat) c) L{*} = Z => 43 = ['{ el let F(S) = Ś => ["{ F (334= ['{ $} = 1 = f(t) By the second shifting the onem, we have T'{zas, F($)% = f(-a). 1(-a) !.11fé? $y = f(t-1) 164-1) = 1. u(t-1) % = u(t-1) //