Question

(k) rep + 3y = sineoso (1) *+ (tan z)y = cos? :, -1/2<< <7/2 (m) cu ap+ 24 = 1 -220 (n) er + (x + 1)y = x, y(In 2) = 1, 7>0 (0) (0 + un - 262+ + 2) = f =>-1, v0 = 5

Answer 1

Try the given differential equation is ad + 2y = simt ,0130 ) Dividing equation (1) by , we have en y = fopen they equation (2) is a frest ander lineal differem troll equcetion of the tube de + Pyz @ - (3) Combaling equation (2) with the equation (3) We have A-3, Q = sehen Hence P and Q ale functions of a Integrating factor (I.F.) = op de = za doga = pdagur, { boggly = my allogg? Eva Pi eloga 202 .: (IPA) = quo 4

solution of equation (I) is given by y ( I-fo) = Q (I-F) dan + c whole cie aebitealy constant. * you as piny 93d7 + Es sive data no yote = - Coletta Ol y = [Co-2014c) which is a solution of the given differential cancel ron () (f) dy + ( tance) y = cosi 5) where I can Equation is is a frest Oeder lineal differential equation of the type of equation (2) Whole P and Q are functions of t. Compaling equation (sy with the equaction (3), we have p= tant, a = Case intertaling factor (Ifo) = pS Pepe = estanada - olog secil

.: Info = pelog Seco = Socal leeloga =97 we have Info = secco solution of equation ist is given by y (IiFo) = SQ 1 I•Fold + c . Where cis arbitraly constant. .. ylheca) = S(coge u feca) da + c = fasa date, sau Cost fect=1} — 9імя +с niy= ser l Pinatci ol y = .00371-3144+ C) which is a solution of the given differential equation 15). im) a do ty=1 - 16) whole ajo Dividing equation (6) by a, we have ovely to y= A - 7) equation (7) is a frest older lineal differentfal oquation of the type of equation (3). L

= pp logue where P and of are functions of an la Compaling equation (7) with the equation (3), we have PER Q = 1 ) integrating factor (I.F.) = = esa dan = e logope , Dogom = m fogo?? = y, { : Blog! 2013 oi IoFo = qe Complete solution of question (6) is given by a y(DF) = SQ (I-Fold+C where c. is albitfaly constanta :: you = Sf ) (ory do +0 =51a-1 dato you = ( -a) to + ya -)+c] of y=1 - 4) + 1

Hence y=( 6 - 12 tu is a (1) solution of the given differential equation (of where c 13 árbitroly Coustento in adly +BX+ILY=4 -18) Whele yulma) =1- 19 Diurding equation (8) by of, we have + (1+ y=1 4110 equation (10) is a frest older lineal differential equation of the type of equation (123) where p cind a lele functions of yo Compaling equation (10) with the equation (3) we have P= (175) and cari integrating factor (ITS) – J'Poloe = = = N(+) de plat hogy) yo fogue = e , polegy 42 - IF's get Complete solution of equation (8) is given by

y (IR) = S Q (Ifo)da + c ( =yqaeyj =S (17(429) data = loetela tc intograting by palts, taking of as frest function and eq as second function, . We have yleep) = 0 en-suedato xea - serdate ...suere9+ -igger = expo-1) + c =y= obt {2016-1)+c} = 11-6) + fete as y= (1-5) tibét (11) . butting a= 'Ina, we have Glomex) = fl-eti) t e dhe * j = 1- Me + same echey ? 30=-cha tema ar? 38: ethy=a3 or-tha +(37) enero

→Oz wehe 1-1 + $) = 0 = -17 @ →ç = 1 > Ć=2 butting c=a in equation (11), we have Y = 11-14) + 2 et which is a Solution of the giver differential equation (8). Note: As been chogy instructions, I have salved first fout balts (k), (I), (m) and (!) of the given five different baltso