Question

Af(t) Use the method of Laplace transforms and the accompanying proof results to solve the initial value problem. y" +9y' + 18y = f(t): y(0) = 0, y'(0) = 0 Here, f(t) is the periodic function defined in the graph to the right. . -- -- - - 1 Click the icon to review the results of a proof. 20 10 15 Square wave Choose the correct answer to the initial value problem below. 1 O A. y(t) = (1 + e -5t)t(t + 3/(t+6) O B. y(t) = 1 18 (-1)"(1-2-316-50)) ?u(t = 5n) n = 0 0 C. y(t) = n=0 - E (1-2-34- 10n)?ult– 100) Te (1-2-342 OD. y(t) = 13 (1 - 4

Answer 1

if f(t+7)= f(t) ; fct) is function with Period 7 L (for) = S f (t) e e) est do пт T 21 est s f(t) est de + f f (t) et dt - - / flt) et du (Ant T ni { . fet) est du (n-177 ntz' let t = kt(n-1) T =) dt = dk I fac f ( k + (n-1)T) = f(K) t= (n-1 T =) K=0, t=nT =) k=T. T - Sk - (n-1) dk so, 2(f(t)) i sf (k+ (0-1)+). e n=1 D

h(fit)) = § 3-1-137 Jfck) est de =) e (n-1 IST s fit) est dr. E Stechestra at alesa Given D.E y"+9y't lsy = f(t); Y Lo=0, y'Coleo let =1 2(y(t) = Y(s) Lly' (t)) = Sycs) -y (0) = Syu) Cy"(t)) SC Sys) - 410)) - YL0) Šy(s) fit) liven t 5 10 I oft Ls - fcel with Period - 5. talo T-10

Lifet) = s fit). est at So, e 106 }, estar (so) (1-25)(1+253) -st -53 ) lipess) (le (less (less) (itess) 1 X slitets — Now [(y") +9 L(41)+L(y)= a (f(t)) - (5² + 96+ 18) yer ve { (-1) Sasn Slitess) role, Items YCS) {(1) - SAS s(3²+95 +18) - 3+95 tis 3² +65 +35+ 18 = S(+6) +3(346) (5+3)(3+6)

rok А B с + St3 S St6 si5+95 +18) A (²+95 +18) +B(32+65) + c(²+35) - s(32+95t is) (A+B+C) 3 + 9A +6B +32) + 18 A s(8²299+16) 18A = 1 =) | By comparing : To =) 68 +36= - 9A+GB+36=0 2 AFB tcro ab(B+C = is. + -3c= -1 2 =) (= ! 18 こ 185 9 (5+3) 18(5+6) S(+95 +18) 2 + - 1 3 [ :] 3+3 stb So, y(s) = ins] [ Y (S) = 1 { (-1) eSA + 2 443 s+ 6 18 0-0

2 + st3 sto Y(S) = [ { tiesne ( nothy 2 ( uct-c). f(t-1)) = é es d (fets) -) I'le's UA CH)) - uct-c).fct -c) se , ellys)- to co d' (65" (3 - Rand)) t 1 Here t (f (t)) = ! 2. S+3 Stb -37 -bt 1- Qe te fit = (1-23+)? 2) So y(t) = ! [ CIuctösn). f (tösn)] 18 YLt) = I [ { [1-23 (4-50 J. ult-sn) -) 18 no option B