Question

(1 point) Consider the following initial value problem: y" +9y (st, o<t<8 y(0) = 0, '(0) = 0 132, ?> 8 Using Y for the Laplace transform of y(t), i.e., Y = C{y(t)} find the equation you get by taking the Laplace transform of the differential equation and solve for Y(8)

Answer 1

0 yutay = gut, o st=8 y (o)=0, y'(o)=0 32 t 78 니 S2 L) - Slcy)-yco) SYCS)- y (0) L(): st4) S 40) -3'co) = sºYCS) – sạco) -3'co). case-1: yll tay= ut ; ost 28 So, say(s)-sy(o) -y (0) + 9 Y() = 4 or, sty(s) -58 y.) - stylo! + 951 ys) –4=0 or, Y(s) [s" + 9 52] = 4 or, Y05) -(40s) [se] Case-2 : yll +9y=32 ; t 78 So, s²y (s) - syco)-y'(o) tqY(s) = 32 08, s3 y(s) - syco) - sy'co) +95Y(S) = 32 or s3Y(s) - 5² (0) - S(0) +95Y (S) =32 or, y(s)[3+95= 32 32 s3+95 S(5249). As we know YCS) = L {y (t)} Hence when y" tay = 4t; Osts 8 then L{y(t)} = s?(s2+9) similarly when yll+qy=32; +78 then L{y(t)} = 32 (5 +9) s 4