Question

10 sin 2t if 0 <t< 4. (a) Let r(t) if t > T Show that the Laplace transform of r(t) is L(r) 20(1 - e - e-78) 32 + 4 (b) Find the inverse Laplace transform of each of the following functions: s – 3 S2 + 2s + 2 20 ii. (52 + 4)(52 + 25 + 2) 20e-S ini. (s2 + 4)(52 + 25 + 2) (c) Solve the following initial value problem for a damped mass-spring system acted upon by the sinusoidal force r(t): y" + 2y' + 2y = r(t), Y(0) = 1, y (0) = -5. Scosinat if oct r(t) = {cosinat it on al = loult) sin(26)-loult-T) sin(26

Answer 1

O<tr 4) a) Given r(t) = En. = { lesina, 5. £ 6«c) = Se**rcode st e losinat dt slo "Sin at dt het I- -st "Sinat det = | Sin 2t = O + + 2 Sinat dt - 4 / 1 = [*] = (1+) I = B (-es) (1-e-sh) So I (r(t)) = 10 -(-e-Ms) - 2 > I = 574 2. 374 20 3+4 -(1-eh) b) is ["{ s-3 ş²+24+2 3 = 1 E'le 3+1 (8+1+1 -40 5 (+157,} = e Cost-4e-tsint

ii) Let 20 = (As+B) (572x + 2) + (CAD) (174) Let s=2i, then, 20 = (2iA+B) 64+41+2) = 20 = -8A-20) + (-44+41) So 4B-4A=0 => BFA de -8A-26=20 33-100=20 (:: B=A) > A=-2 So B = -2. Let s=o, then 20 = 2B +4D = - 4+4D D- 20+4 = 6. 4 het s=1, then 20.= (A+B) .5+ (C+D) - 5 > A+B+ G +D = 20 S = 4 -2-2+C+6=4 > C = 4+2+2-6 = 2. so L {erua Car ava 94] -f"{2} ++"stan) =1"{{25-3)(**20* 2) +(25+9) (974) (874) (05+23+2) (6+) +46 'Lentvar) Sin 2t + 2etcost +4e-tsint. = -2. Cosat -

2oens u(t-17) f (t-1), where (574) (5+23+2) art) is the unid step function and ["{13746572872 +46-(tan) f(t) = So f(t) =-2 Cas2t - Sin 2t+26*6st+se+sint (by ci) So f(t-1) = -2 Cos 2 (4-0) – Sin 2Ct -r) +244-r) cos (t-n) Sin (t-11) =-2632+ - Sin 2t temat + [2cost-4 4 sint] So, moru 5+23+2) -ut-n) |–2 Coszt – Sin 2t + e^-+ (-2cost – 4sint) Given equation, Y"'+2y'+ 2y = r(t), with Yoo) = 1 4'0) = So [{y"] +2 [{*}+2447}= [ (764)] = $y (3) - SY0-y'(0) +2sy (1) – 2y (0)+ 2y (0) = 374 = $71%) -3+5 + 2sy(4) - 2 + 2y(1) = 2011-es) 574 369726+2)y (3) s) = 20(1-ena) 874 20 (1-6-its) s-3 7(a)= (+2+2 3723+2 =-5 20 (1-ets. +5-3 + >y (t)= Clementerian Agamemn] (874) (25+23+2) s-3 3+25+2 y (t) =-2 Cos 2t - Sin2t + 2e + Cost +4 6+ sint Ecos 26+ Sin 2t + en-t [2Cost +4 sin t]}u(t-1) tetcost a4 et sint

> y(t) = u(t-sw) [2005 2t + Sin 2t +e"-t (26est +4 Sint) -26s2t -sinat +3 ét cost.