Question

Find the solution of the following Initial Value Problem by using the Laplace Transform. In your answers, always write y(t) or Y(s), not just y or Y. If you need a Heaviside function, write U(t) or U(t-a). y"(t) – 6 y'(t) + 9 y(t) = S(t-4) y(0) = 2 y'(0) = 3 L(y(t)) = Y(s) (y(t)) = LTY"(t)) = (52 - 6 5 + 9) Y(s) = Set up and solve the partial fraction decomposition of y(s) Y(s) = 2^3 + 65232 + 15–332 245 where A= B = C C=D The solution is y(t) =

Answer 1

Given that g!)-6jt) Hay 19). Sle-y) yo) = 2 fyty}v= Y(S) y10) = 3 {ytoj r= 5xls)-yl) – 0 2{y":)) = 32415) -1)-910) - from SYG) - 9 PM LP348)} = 5Y10-2 froma -fy"14] - 33465)–s(2) – (3) if y(t)} 2.5$)-25-3

Lolt-u)-86 [52415)-25-3] 46 syis)-a] 49% (3 šy(s)-25-36576) +18 #Mis) 6-48 (3) [s? 6549] -25+9 = ells Lys) [s265 +9] = 845 +25-91 Y(s) = 645 + 25 29 646 425-9 (52 65+9) 75 -3)2 (2) A + B Hit o els 53 1533 6332 e 45 +25-9 = A (53) +B + ce 4s & 4S 425-9 = AS-3A + B tice us Compare like forms

26 = 8A Deus = ce us c=1 poo :-3A+B = -9 Hege -3(2) + B = -9 - 6+B=-9 B =-946 18=-3) Applying laplace inverse

: l' } =zett OC iya) - zesttest3t+(ty) Wien

THANK YOU STAY SAFE

PLEASE ASK IF YOU ARE HAVING ANY DOUBT

PLEASE UPVOTE IF U LIKE THE WORK