Question

A particular solution of the equation y" + 16 y = 241 + 2 sin(4 t) should have the form: ae4l+ct sin(4 t) +et cos(4t) ett + c sin(4 t) + e cos(4 t) a e^tt+ ct sin(4 t) a e"! + c sin(4t)

differential equations

Answer 1

page I, given diff. equation is yu+164 = e 4t +25in (4t) to find particular soln auxiliary ean in the is FCDI = D2 +16D we know form of op exator we know w I e eat - - cil FOOD Fra) = and - sinat FCD2) (ii) sinat FC-a2) particulas solnis Teut) + 1. 2014) D2 + 160 D2+160 - eut + 1.2 Sinut / From (1) ( -16 + 16D (4)2 + 1614) 8 (1) e 4t Sint 80 - 16(D-1) - 80 eat ta (O+1) Singt 16 (0-1) (0+1) e4t (4Cosutt SINGE) 80Ă D2-1 - e4t sinut till Cosut + + 5 (021) ID 2-1 e 4t 80 1 I (COS4L) Sinut) 2 -16-1 8 (-16-1) - eat t cosut tsinut 80 - 34 - 1 3 6

page 2 so particulos son is = a e4t +c81n142) te cosult) " correct option is option (b)