Question

Page 4 IV. (10) Use the Laplace transform to solve the IVP y" - 2y + y = f(t), y(0) = 1, 7(0) = 1, where t<3 f(t) = t-3, t3 You may use the partial fraction decomposition 70-28+1) -1,2 = (+*++* - , but you need to show all the steps needed to arrive to the expression (+28+1) in order to receive credit.

Answer 1

Qooles criven the IVP is y-2y' tY = Ft Yo=1, 40) ) where, Ft) = o, t <3 1 Festa Taking Laplace transform on both sides of we have since, f(t) at L?Y} – 2 L34'} + L343 = L}F(t)} -3 (2184) --Sye) –Y'03] -=[54443 – Yo] +LM1 = L*u)? [ Li --S.1.-1] -2[SLY} ---J + L347 = L{FC} SLM – 9.-1 -- 25L193+2 +1143 = 100 7 (92-25 +1) L34 ) –S += L{F)} O mes 2390) = fet - LJUB = est suhde ser fe de - L1F4) )= past.odt + (It) 20 443) + $$(43) de => Law)= (+3) pestklepas bestellt 9 1.60) = (423) 641-10-2).ended est. (4-3) dt 2 LTH)=

- Elf(t)) = -1 Slim (t-3) too est L?50) + (0-9]++ le est ilim (t-3) is a form too Then By (Hospital Rule lim d(+-3) too वर dest) It to sest=1=0 2 La2} " L/FH)) » 23612 늘 [em_728] Ž[o-ess 32 in ②, Case have using (8-25 +1) L293 – 5+1 = 735 s2 +(S-1) = (S-13² L33} = 235 5² 4393 = s215+)2 e35 + (54) ē 35 Taking L39] = 5²1572 (St) Inverse Laplace transform on both sides, we Y(t) = 315121 have

Now, let F(S) = e 35 ² (8-12 - L3 F(S)} = ( +835 3) 173 F(s)}= f(4-3) U(4-3) Also find gello F (1-3) let f(t) = cos? S (5-1)² 39 ft) = 13 0932+ stk 6-1)} 7) FU) = 136 +20'] }}+1473-213 co? +² ezet the ( In question, partial fraction) given = F(t) = et tai +2th 12- 11-1 (24 1- + in- + 2 1 ["38_28? (nt and L" } ph} = > F(t) = tet + 2 -2 et ..L= and Lo=1 F14-3) = (3) t-3)+2 +143)-2ell-3) and t-t=1 a) f(43) test 3)-34-3)*+--2643 13) 7 F (+ 3) = tet tt-l Then equation. becomes. {"3F18)3=436-43 1 = + + 3) -564-37 t- Using & in we have YE) = [t&#9_se +t-it et. th pockenheim 1- 2) YU 2 tet-3 - set 3) +t-1 & et (that=1 V 17 slo1 (y) = t_set" + e + + - which is required solution of above IV PO
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