Question

Solve the differential equation: Y" + y = sec(x) given that the complimentary solution is ye(s) = ci cos(x) + c2 sin(x)

Answer 1

Solving the differential equation we get
The given differential equation is y"+y = Seeln) Let y = Aemn (A&O) be the trial solution of the equation 2. y'= du a au (Aemny »y'= Amen and yl= dul = dan (Amemang & yll= Amenn putting the value of y" and ly in @ we get auxilary solution themnao [m2+ 1] =0 m2 +150 [since Aenzo Am²e mu y Aemn 3 m2 = -1 »m=tfi 3m. Eti are The complementary function is Ye - eosin) + C, Sinen), where a and arbitary constant. For P.I:- - Cosy), Let Y, and y=sinly) Now, wly, 92) = 9, 92 ५, ७,

Coslr) sin(n) 11 -sinin Costw = cos(n) + sineen) = 10 ^y, and y, are linearly independent. Let yp = 9, (n) (4) +90) U₂ (1) Where, Ulu Where Rim) 2 Seeing du لا را E-S Sinen seeing 'du 1 Sino) -dn Costro -s tamcn) an (- Inicosmy 1) in I coser) and U () = 5, (M) (1) Wiggy dn CoS :Seew) 'du 1 - COS Cos(u) du = s dn = n.

i. Yp=4, (4) 4 (4) + 4,142 (4) * Yp = cos(n). Ini cos(WI+ sining.n. ". The general solution is y = y + Yp y G COS(M) tesinlus + cos(a) en leoscu) Insinly) solution of ☺ which is the required