Question

Use variation of parameters:

4. (D^2 + 2D +2)y = (e^-x) csc(x)

Answer 1

112 9 The given differential equation is (B? +2D+afy = ē y coseclor) - 1 Where D = doc Egyceffor (iis a linear differential equation with constant coefficients. co Aurillóoly equation of (1) is given by metam te=0 - tel Equation (al is a quadratic equation of the type aimet bm+c=0 31 Comparing equations (2) and 13), we have - a=l, b=2 and C=2 to the tools of the equation (p) are given by m= -btJ62-496 20 > m= -2 + 3=4xl x2 우 na I s 408 - = 2tJ-4 = -21 Jyre {=0 = -21% = -15% 3 we have m = 72 = XL2B, where d=-1 and BET

.: The complimeutty function (CoFJ PJ of equation (1) is given by CoF = Ora la Cos B4+ą SMBA) * = ēl ci conat Disa), 6:2=-1,B=1? . CoFe = ē ( Coslav) +691 (0) — 14) of Gofo = ci ēC08(1) + Get Sinfa) 759 of Cofi = Ciyit Goya — 161 Where Si= ē Costa) - 17 and y= ēt sinla) - 18 Using variation of balauter method, det Y = Ayit Byafaj be the particulat solution of equation (1) Whele A and B are functious of ot and are given by A = 1-92) X do Grlod and B= 5 yix de 3 where X = ēt cosecco) —112), the function on the light hand side of equation (I). W

13) functious and w=wrous kian of the - S, and yo na w=1 - yi you W= le coura) ēt since 1 ē99-ecospa Cosa-ētsino =ē Cestayf ēCasa - Sin(a)? - siulo - SM19) -ē Celle)? = @ 79 coglie-e gjy(3) Cos(a) te pt ginna) tēts sinto) C9 (9) = ē ? Cosple) +ē 29 314?(9) = ēna (Cosplt) + 31 (+)) =ē pfx " w = ē at 113) Now from equations (8), (lo), clay and 1199 we have A = r le siyo ēY Cosecca) do we have have A = 20

S 2 A = 5 (-es) gulay cosecca) de les =-347, 310 Coseco = 13 so we have A = -4 - 14 Again from 17), (11), (1.2) and (13), we have 3 = S leucosin) (e casecfu) da Es pot cases cagecle) de = cos(9) Cosecleda = { costal de = Scotlanda ,Soz Coseco = sho = plog Single pand Coto = Corso .: B = log SM(+) — 15 Now filom (7), (8), 197, 114) and (15), we thave yb= (-1( ēCoasty) +(abog sir)) lēt sin(a)) --Yp=&pf=% Colla) + Siapa Jogasinan

1900 Hence we have Yb= ē% - 0 Cosa) + sin(a) clog S4 (213—116) Is the particular sodation of the given differential equation (I). Now from equations (15) and (16), we have y = CoFo typ 82 y=e(Cesta) + Bin (-1) tētf-Costa+-Din (2) Jog 3in1)? y = ē ||c, Costa) +G_99412) '+sca Cosine) + siula) Jog Sin(a)} | is the general Solution of the given differential equation (1). where ci and Q ale arbitraly Constants