Question

13) (15 pts) Solve the given IVP. y" + 2y' + 2y = 10 sin(2t), y(0) = 1, y'(0) = 0

Answer 1

Goren y" + 2y + 2y = lo so(2k), 9(0)21 4(0)2 homogeneous Soulon y"+2y' + 2y =0 0+ 20+ 2.20 YLL); cēt cos (l) + czēt sinch) Particular solution Yochte A Cos(2+)' + B Sın(24) y = -24 8in (24) + 28 cos(24) y" : . -4A COS(2+) - 48 39(2+) -4A COS (24) -48 Sinc2t) + 26-2A Sin(24) + 28 Cos (2+)) + 2 (Acos(27) + B SinQt)) = lo sincet) cos(21) (-44 +48 +28] + 8m(2L) {-48-49 428] = 108190) 48-2A 20 = 24 zu8 21 A2 28) -4A-28 = 10 -4A-BI 2 10 -5A=10

YpCk) = -2 Cos (2k) - I sin(24) 9(4) 2 cēt cosch) + Gé' sinch) -2 co$(24) – Sonce+). 46)-1 I= .0; -2. s'e: c[-écoscey -e'since] + cz [-êt surces a el costs] + 4 Srnc2t) - 2 cos(24) O = yoz. o :-(;+Q-2 C2 = 6+2 = 5 19CH) = ze coscie) + së'sinck) COS(21) - Sin(24)
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