Question

please with clear font

this is a differential equation question please solve them with steps and details also make sure it is correct 3. (6 points) Solve the IVP: Y" +9y = e3+ + 8(t - 27),y(0) = y(0) = 0.

Answer 1

yutay = 83t+s ( ta) and; f(0) = f'(o)50 we will solving given diffe ean by laplace transformation Lry") + L gyl=hrest) + L [s(t-2a)] (5²418)-sy(o)-ylko))+ 9 y(s) = 1 teks (5-3) I since; Lreat/= & L{st-c)]=ēcs S2y(s) + gy(s) = 1 (5-3) teas yess (52%) = bergtoes (529) yes) (s?tg) =/ I thas (S-3) + ets (S-3) (549) (5+9) taking Laplace Inverse [y(s)) = 1 S-3)(52+9) th s²tg

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1 509) = 0 (54] El= Isml3t) 3 la sf(s) = utt-2a) fit-24) &mu Llesfcs)] = ult-chit-c where alt-c) heaviside function Lors S²g 3 altad) | Sm[3 (t-2x) 111 3:) = 2 ( 693se-ty) +41 () So, Ilt) at 18 et 1 CUR(E) I Smit 18 18 + lll-2x) Sm/3lt-2k)]