Question

Consider the following IVP y″ + 5y′ + y  =   f (t),     y(0)  =  3,    y′(0)  =  0, where f (t)  = { 8 0  ≤  t  ≤  2π cos(7t) t  >  2π

(a)	Find the Laplace transform F(s)  =  ℒ { f (t)} of  f (t).

(b)

Find the Laplace transform Y(s)  =  ℒ {y(t)} of the solution y(t) of the above IVP. Consider the following IVP y" + 5y' + y = f(t), y(0) = 3, y'(0) = 0, where f(t) {85(70) > 2.5 Osts 20 cos(7t) t > 2 L {f(t)} of f(t). (a) Find the Laplace transform F(s) (b) Find the Laplace transform Y(s) IVP. = L {y(t)} of the solution y(t) of the above

Answer 1

Given IvP is y'+5y' ty=flt) Y(0) = 3, yllo) = 0 and f(t) = 8,osts21 Coo 7t, t>271 { f(t) can be written as $4+) = 8 (u(to) - ult-2)] + Cogit [ult-270)] where ult-a)= tsa t<a unit step junction Now L[114)3=L[8ult)-864-28) + Costult-20) =8L [ult) ] -81 [ult-291)] +L[cost ul-2)] Ao L[ulta)] = è as A L[ cosat] = zna By second-shifting theorem L[ult-a) toobt] = pass 52+62 S Sta²

:h[uct)] = eos = 1 L[ultan)] = 2 L[657t] =5 diyo 52449 L[coolt a(4-27) e2TS s 52+ug [[+1)] = 8(1)-9 1820s)+ e 2n's Ehtes L[f(t)= 861-82901) teams (tyg 6h b) yl! +Sy'ty - H Taking haplace transform L[yll +Sy'+y] = L[+(+)] L[y] +5l[y'] +L[y]= [[+(t)] +(82Ys) --sy(0) -Y'(0)) +5(sYes)-y(0)) + %18) = 861-8205) teatro (sitra) Shtz

(52 Ys)-35-1)+5 (sYO-3) + (5) = 861-829s) teins (5 149) = (52 +55+1) kis) - 35-15= 861-62ns) tens (vo) =(52+55+1) YCS) = 23+35 - 8 € - RTS terms ( cuya) 8 ents -2005 7 YIS) - . 35+23 5450+) e + é s²15+1 (S²+SS+1) (52+49) Hence, - 2nS + e S²+55H S²+55+1 le S 5?+49 8
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