Question

ا = (هر را = (0) - (a) = ) = (ه) و = (الا ( 3 / 2 2 و و 2 گیا - ۔ 41e) = 4(e) = 0 24+y = fext @ 4

use the convolution theorem to solve the I.V.P.

Answer 1

y ty= 3 fins 9(a)= y' (6) = 0 transfrom both sides Talets Laplace L2 yll (1 9 + Lycurga 3. L2 flash 12 YN-14 (6) - 4'(o) + 37181-4 (0) = 3. F(d) p24(0)-orotllo = 3 FIN (12+) YIN = 3 FIN Yl)= 3 FIN) srsti 1/+1 Anal) + BS 1= =) A =/ 2 1 A (uti) +b sary 120+ Bl-1) =) (Bani) Enes a city YA) (2 - 2 + 1 ) Fun HA= 3 - 3 d h(t) = L-IL HINY L2 youY= HA ELS) -1 906) - thx A) ) HIME / - /sty L-I (HINI = LILW-3 (1(+1) = 3.1-3. ett. (/) = 3-3.0th ya h* f) ist) . y) = 1 10-2ệt). f (x-T).dt

(O) 1 y"+241+ y = fun og(u) = Lagenya FIA), let Talats lablace transform of eqr o both sides 2 y(x)) + 2 L2 y' cury & Lycup = LL fluss s² Yin) - sy (o)- y' (a) +2 (171) - 4(6)) +Y() = FIN 82 yrs) - s. 1-1 +2(d. 1680-1) +7(4) = F(d) p²Y/H) asal +2 s/d) - 2 + 1 (A) = f(d) +3) YA [ 12 + 2) + 1] - J-3= F(S) Yl= f(a) + (8+2+1) 12+2+1) Talcing inverse Laplace bothside L'idyranga -1 FU). - ) + til Set It?) (1+12) ya) = (-1 (BIF ) + Lil 'st) + 2.2 (J + y 2 ) yota Lall .FM) + 2 F) te (4) +2.0*2-1(2) cet - 2 = Huh, when the = L-1 (HCA) 12 ti ... 90k)=HW.Fun) + él. 1+02.exx You = -1 HA). LA (Full + et trett ..:(-1 (HAI= L-1 (4+1/2) = ett you at tiet. f (x-7).dt tetraktet t et fantlidt tehtanen ya) = (

y" + 2ylty = fone, O Y (0) =y! (a) = Taking Caplace transfrom both sides Lqylcuis + a Layllary + Lycay = LL Hary 12 // Al-Sy(o)-yllo) +2 ( 84A1 - 4(o)) + Y'AI spyro-0 +2 (140-0) +461)= F(A) you l 12 +2141] = FM) LLYMy = yen a I . FAI FAI 1+12 let (f2' LM (1 ) HCADE - - Lullam = L ( +12) = B hlula ex.x. 24 (019 = HINNOFIN) =) Yeni= (h*f) (a) y cal= (h&f) (x) yokl= 6" ēt z. f(x-2).dt