Question

Find the solution of the given IVP y" + 3y' + 2y = Uz(t); y(0) = 0, y'(0) = 1 + e-(t+2) e-2(t+2) + e 2 a. y=et-e-t + uz(t) [+ b. y=et +e-+ + uz(t) [ – e-(6-2) + že=2(t-2)] c. y = e-t-e-2t + uz(t) (2) - e-(4-2) + že=2(t-2)] + d. None of these

Answer 1

r25 l Y" + 3yl+2y= U2lt) y lo) 0) =0, y los = 1 Taking Laplace trams, form, in (sºyages Yvo) - g'oy) +3 (syes ) - 1103)+2:4653 -> (5435.72) yes) -5.0-1-3.0 = e (5+1) (5+29715) I StV (S+2) -25 e2s +1 s € 25 T Yes? + 2 SCS t) (5+2) Now, let с + + B St SCS+) CS+2) S S+2 2 powers a of swe have A +B+C = => A (S+1)(5+2) + B SCS+2) + CS(5+12=1 => (+ B)5+ A +23+6) 5+ 24 = 1 How comparing like co 3 A +2 Btc 2 A 1 yA=%, B+c=-1/1 => A= (2 Bte) - (B+C) = -2/+th A= Ba-), Ca ½ / 2 Btca 3/2 >> į :. SCS +) CS+2) -19 10 t sti I 245+2) ... From @. e 25 -25 25 sti - Ie-25 T 5+) ୧୮ 25 f ezs sti 2C5+2) You a 0 25 (5+2)-(5+1) 275+23 S+ 1)(8+22 sta Taking inverse Caplace transform YHI ; 2t+4 Uzlt) =et e 2t + c Ų It) Ult) T1 - oct-2) thy er 2(t-2) Thus & is the answer. est e-2t + 2 Ult) - Eft? U2(E) + = e į e