Question

1. Solve the following differential equations: a. xy'=y+Vxy x+2y+3 y'= b. 2x – y +5 x+2y+3 y'= x+2y+5 y cos(x+y)+x+y d. sin(x + y) + y cos(x+y)+x+y C. y'=

Answer 1

". The equation reduces to dyr s'tay' du' 24-y! let us put y'aven' Til a vruldre : vtn'de dul dili n't anul an'- un' 1+2u n' dhe an 1+2re arv Itxare ² 2. v R-u luz de dul dhe I. 20 Tara dre both side ar, 24102 che Integrering we get. 2 sam- v- & leg (14222) alga't loge. where Lage is constant. an, a tan al no y (1+ ar a tana x+1% [ezt , -kalyan egy) ergen? mylvs] 2 lag (1+(377 algen' 2 tan al syti su+13 5u413

in homogenous form, let us Mt2 y + 3 ut2 y of 5 To reduce this pur non'th, y = yk. m't24' + h+26+3 ulday't h+2kts let choose nu such that ht2ur +3=0 haak & 9 20 which is ! The solution does not exist. di a not possible d EYCOS (nty) at (nty) sin(n+7) + 4 cas (n+7) A (mt). ut us put nty 1 te dre d == (-u) cose te 3 an dhe sin utv-a) care tre - (van) care are du +1. sinet (van) carto -(V-u) casse-p & sino +(V-u) care + x sinut (von) cortv. sinu sinnea (van) care te - sinive du at [sinu+ (van) care to do ao m du & Nav = o. Maasing and N=sinot (vn) car tu. an, an where am =-core an am - - cave am que a ar : The egun B enact. :. The primitive of the equno is s-sinne au + f[ sinu + CU-n) care to] che = o. is consider as constom for the 1st interrey 2nd cast & core + vesino & car - usine + y² : (htt) sin(mt) & Cas(ut 4) – asin (mt) + (uma)

". The equation reduces to dyr s'tay' du' 24-y! let us put y'aven' Til a vruldre : vtn'de dul dili n't anul an'- un' 1+2u n' dhe an 1+2re arv Itxare ² 2. v R-u luz de dul dhe I. 20 Tara dre both side ar, 24102 che Integrering we get. 2 sam- v- & leg (14222) alga't loge. where Lage is constant. an, a tan al no y (1+ ar a tana x+1% [ezt , -kalyan egy) ergen? mylvs] 2 lag (1+(377 algen' 2 tan al syti su+13 5u413

1. x4l ya unya as, y'- Ytury Y 19. - vtu vetu vn & Uhinen vtu unture 1. Le of Man dre an, an Integrating both side we get. constant. 2 Vs & a legna lege where lege is goo.logen. yz Log ind ay, poa Ye logen. rly cogen [ yaren] =tru (b) du = ut2y+3 2n-1+5. To reduce these this to the homogeneus form, let us pur x=rith, Yay'tk n't2y'+h+2K+ 3 2u'ryl +2h-k+5 Let y choose hik such that. htak + 3 = 0. ah-k +5 -0. These give hot + 2 x W we get. 5h = -13. k = -2x + ..h=-1/ -26 + 5 = 15 U