Question

For the differential equation d (+2y) = x() (a) Determine the homogeneous solution. (b) Determine a particular solution when the input signal is x(t)=12* cos(4t)u(t). (c) Determine the complete solution with the initial condition y(0)=0.

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Answer 1

dylt) + 2y (t) = xlt), given differential equation dt homogeneous solution in order to bind out homogeneous solution mare selts=0 dt det ylt) + 2y(t) = 0 (Dtz) ylts=0 (6+2) 5 (4)=0 4) ) (t)=keat 2t y(t) = C, e et ult) D Particular solution for the input siand (t) = 12 13 "Cosht4lt) dyt) + 24(+) = 12 est cos ut ult) dt y Ct) +24 (t) = 12 est casut alt) The porticale solution is in the foren Ip' SPF A est cost & Best sinut = 4A est (-sinut) + (3) (A) est cos 42+ -3 Be3 sinutt Best (4) Colat Cosat ( 4 Be 3t 34 e 3t) + sinkt (-3Best 4 (-4A) 234) est coskt (4B-3A) sinut ( 3B+4 A) e 3t we susslitate equation ③ in ① Gut et cesht (4B-3A )- sinst ett (4A43B) + 2 {A 2 osht + Beltsinit) Iz est cost

est cos 4t [ 4B-BA +2A] - et sinnt [49436 +2B] = les court est cost [4B-A] – est sinkt [4A + 5B] = 12 & 3 cos4t compading both sides 4B-A = 12 solving equations - (4A+5B) = 0 -# +63 = 12x4 HA+SB= 0 21B = 48 B=48 21 4 x 48 - 12= A 21 20 sol = -20 e 3t cost +48 e sinut Yp=Pastiglas solution=-2002 Cosett 21 48 est sinkt yp= 23 (-20% casut 48 sinut 21 complete solution with yol=0 y y(t) = homogeneous Particulah solution solution Ilt= ser c, ezt sp= est {20 (sht + 2 sinut

stt) = (let to [ Colt + L& Sinut ] given Co) - 이 9 디t U ) + 1, -e + 0 q=0 구 5 가 내 20 ezt te 바를 cas42 +48 sinkt complete solution