Transformada de Laplace- Resolução de Problemas de Valor Inicial

Use a transformada de Laplace para resolver os seguintes problemas de valor inicial a y 4y 4y et y0 0 y0 1 b y y cost y0 0 y0 1 c y y 2y t y0 0 y0 0 d y y 2y 5et sen2t y0 1 y0 0 a y 4y 4y et y0 0 y0 1 Ly 4y 4y Let Ly 4 Ly 4 Ly 1s1 s2 Ys s y00 y0 4 s Ys y00 4Ys 1s1 s2 Ys 1 4s Ys 4Ys 1s1 Ys s2 4s 4 1 1s1 Ys s2 4s 4 1s1 1 1 s 1s1 s 2s 1 Ys s 2s 1s2 4s 4 s2 4s 4 s 2s 2 s 22 Ys s 2s 1s 22 1s 1s 2 As 1 Bs 2 yt L1Ys L1As 1 Bs 2 A lim s 1 s 1 1s 1s 2 11 B lim s 2 s 2 1s 1s 2 11 yt L11s 1 L11s 2 et e2t yt et e2t b y y cos t y0 0 y0 1 Ly y Lcos t Ly Ly s2 Ys s y00 y0 Ys 5s2 12 Ys s2 1 1 5s2 1 Ys s2 1 5s2 1 1 5 s2 1s2 1 s2 5 1s2 1 Ys s2 5 1s2 1s2 1 s2 5 1s2 12 a1 s a0s2 1 a3 s a2s2 12 s2 5 1s2 12 a1 s a0s2 1 a3 s a2s2 12 s2 5 1 a1 s3 a0 s2 a1 s a0 a3 s a2 s2 5 1 s3 a1 s2 a0 sa1 a3 a2 a0 a1 0 a0 1 a1 a3 1 a3 1 a2 a0 1 a2 1 1 0 a1 0 a0 1 a2 1 a2 0 Ys 0 s 1s2 1 1 s 0s2 12 1s2 1 ss2 12 yt L11s2 1 L1ss2 12 sint I I Fs 1s2 12 ddt L11s2 12 L1 12 213 s2 122 sint cost t 2 ddt sint cost t 2 12 cost sint t cost sint t 2 yt sint t sint2 sint t2 1 c y y 2 y t y0 0 y0 0 Ly Ly 2 Ly Lt s2 Ys y00 s y00 s Ys y00 2 Ys 1s2 s2 Ys s Ys 2 Ys 1s2 Ys s2 s 2 1s2 Ys 1s2 s2 s 2 s2 s 2 s12 1 12 4 1 2 2 1 1 1 8 2 1 3 2 s1 42 2 e s2 22 1 Ys 1 s2 s 2 s 1 A s2 B s C s2 D s1 A lim s0 s2 1 s2 s2 s1 1 21 12 B 1 2 1 dds21 1 s2s1 dds s21 s11 s0 B 1 s22 s11 s21 1s12 s0 022 011 021 012 B 22 11 21 12 122 12 14 12 B 12 14 48 28 28 14 C lim s2 1 s2 s2s1 1 22 21 143 112 lim s1 s1 1s²s2s1 11²12 13 13 Ys 12s² 14s 112s2 13s1 yt L¹12s² L¹14s L¹112s2 L¹13s1 yt 12t Ut4 e2t12 et3 yt t2 Ut4 e2t12 et3 d y y 2y 5etsin2t y0 1 y0 0 Ly y 2y L5etsin2t s²Ys s y0 y0 5Ys y0 2Ys 5 2 s1² 4 s²Ys 31 sYs 1 2Ys 10 s1² 4 Yss² s 2 s 1 10 s1² 4 Ys 10 s1² 4 s 1 10 s² 2s 5 s 1 s² s 2 Ys 10 s1s² 2s 5 s² 2s 5s² s 2 10 s³ 2s² 5s s² 2s 5s² s 2 Ys s³ 3s² 7s 15 s² 2s 5s² s 2 s² s 2 s 1s 2 Ys s³ 3s² 7s 15 s1s2s² 2s 5 2 s2 s² 2s 5 ₂s1 ₃s2 Ys ₁ s² ₁ s ₃ ₂ s² 2s ₂ 5 ₂ 5 ₂ s 2 ₃ s² 2s 5s 1 s² 2s 5s 1s 2 Ys ₁ s ₂s 1s 2 ₂s² 2s 5s 2 ₃s 1s² 2s 5 s 2s 1s² 2s 5 s1 1 3 7 15 ₂1 2 51 2 ₂ 1312 s 2 2³ 32² 72 15 ₃2 12² 22 5 ₃ 13 53 3 52 7 5 15 2 1 5 85 15 2 131252 2 5 55 2 135 153 2 5 5 532 1 34 522 2 1 4 52 354 2 1 53 656 2 2 2 1 34 1 2 1 14 2 2 1 4 3 2 354 2 1 7 53 656 2 2 15 2 54 Ys 14 s 54 52 2 5 5 131215 1 1315 2 Ys 14 5 5 52 2 5 5 131215 1 1315 2 52 2 5 5 5 12 4 5 5 5 1 6 Ys 14 5 1 5 12 4 6 5 12 4 131215 1 1315 2 Yt 14 L15 1 5 12 5 6 L11 5 12 4 1312 L115 1 13 L115 2 yt 14 et cos2t 34 et sin2t 1312 et 13 e2t