Question

Transformations at Work Solve the IVPs in Problem using Laplace transforms. y" + y = cos 3t; y(0) = 1, y'(0) = -||

Please substitute cos 3t with cos 2t instead. Thank you

Answer 1

Given IvP is y"+gy - Cos 27 410)=1 440=-1 Tere laplace transform Lag"(t)} +9 Lsycts} = 24 coszt} 52715) - SY10) –y:o) + 9 YCS) = S S 2+4 (52+9) yes) S +1 = S 5244 sts => (579) YIS) = 5244 S S-1 + => EY(S) = (3²+4) (5²9) 5249 Take invese laploce transform 'Y")) Ľ4 cstes} (5²+4) s tly s 5*)-275603 consider CS+D S AbtB (5²+4) (3²49) (544) (319) → S = CAS+B)(579) + (CS+D) (52+4) = 93 +995 +3+3A +688 +45+05+4) salato 33+(B+D38+ (9 A +41)5 +(98740) comparing both sides > ATC=0 gA+40=1 · 90+40=0 B+D=0 > A=-6 :9A44021 90+4C -1 A='s 3) -50=1 » 05-1 also B=-D 9 B+4D=0 = -90+40=0 = -50-0 => 0-0 - B=0

S Iss rss s2+9 (S2+4) (5²9) 5274 S .., = 5"} r44) 65478] 54352#x}-324 5²+g Š cosat - s dosat also s 1 = Cosat Ik ;] 1"{'shtg} = ģ sidst 2+9 S Lien @ becomes ģ's sinat yrt) - 105 2t -Ścos Bt tios 37 - 5 ما ) ۔ Cosatt cosat - 3 sinat ..y(t) = COS 2+ + 4 Cosat 5 sinat 30 5 This is the required solution.