Question

please solve

Find a particular solution of + 4y = sec(2t).

Answer 1

J: Find a particular Solution of dy + 4-4 = sec (at) Answer: We can find the solution by melbod of variation of parameters Firstly, solve the associated bomogeneous linear equation Yc= ciyit (zYz] dy + 4y = 0 dtz o m + 4 = 0 m²=-4 m= + ai ( Solutions of the foom 4+Bi Complementary Solution where d=o b= 2 Ic=creat cos Btt Czeat sinist .: Yc= ci Cosat + C2 Sin at :. The linearly independent Solutions of the corresporsling i bomogeneous equations are Yicte) = Cos at Yz(t) = sinat Yi' (t) = -25in Qt 42' (t) = a cosat Wronskian W=19 - lyi yal Ws | Cos pt 1-2 Sinat Sinat | 2 Cosat = 2 cosa al + a smat - a (cosatt sin?at) 2 Cosiat + Sinat=1 . W

W 2 = Wielo Sec at Sin at an sinat sec at a cosat ☆ - tan at We = l cos at T-2 Sinat 0 Secat ] 3 cos at x sec at = 1 The particular solution is given by Yp =U9, (E) +V Ya(t)) where u= (Widt and v= (IW z at - Stasheet . u= (W1 dt = (-tanat dt = -7/1 Stanat alt J 2 > issos * Isl Eos atl I lolicos all v=Swaz a dt = Se at a het Particular Solution 9p = UY. () + V 42(t) I lolcosatl. Cosæt t sio 26 -