Question

5. (11 points) Solve the following initial value problem, y" + 3y + 2y = g(t); y(0) = 0, 7(0) = 1/2 where g(t) = 38(t - 1) + uz(t) Type here to search

Answer 1

Solver We have V"' + 3 y' + 2y = 3.561-1+ 0,[+) , Y10) 20, Y'(o) = 16 Taking Laplace transform of both sides, we get L{y" +38'+2y} = {3817-1 + 4/+} = {s:4/11 -s Yeo) - Y'[o]} +31 syist - Yo)} + 241) B = 30 + és or (6++34 + 2)ys) - * - 3et+e 3 è yls) = zato, + 2182734+2) 1²+30+2) Als²+3+2) 210²+31+2) Taking inverse 7141. - Laplace of both sides, we get 3ē9 529 + Als stu1A+2) (A+2) 21s+la+2 (5+2) -0. We know that according to inverse transform rule if I' [Full = flt), the L'{@as Fisi) = flt-a) Velt)

Here 1 - 1 16+1)(4+2)/ entero! =1"/ trata cea mai Il=et- ezt +21 | +1 and 214+2) 2A AA+ 1A+2) - -* + + 1.1.1" stycz) =249, 3400) 4,10 And ie?"IU, +) ē - 24 (+-2) - + 2 - ftė 217-2) e 1111111+2) 2 Thus required solution is yo = 3[ <+1, 3439] 0,147 +7,3149344 10 lucu + 4 let. 804) Formula Used » O L{ $14-037 = eco Zsucks L" Lloret {"+1=1

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