Question

ether they are on you will receive an Ffo 1. Solve the first-order differential equation dy - x2+xy+ya with y(-1) = 1. (10pts) dx 2. Solve the initial value problem dy + y cot x = y sin x, with y(1/2) = 1. (10pts) dx 3. Given the system of linear coun X2

Answer 1

حل ² tayty du se y (-1) = 1 22 This is a homogeneous equation. Put y = Vr. dle (vx) v + x dv da are du do de la x²+xy + y² dhe 22 ze?+ a vx + (x) V te dv dre (1 + v + v?)X? 22 > v tre dv ds 1 + V + V2 du da Ituz d > da x I tva S a te uz s dhe do → tant (1) tlogk loge + log k = logak tan (log ak) x tan (log xx) (: V = 4 / - y n

Now y(1) iie y=1 when x=-1. = -1 tan llog (K) tan log 1-4)) = -1 =) log (-k) tañ (-1) 311 4 3174 -K - e : Solution : 3/4x)) y tan (log &-é tant tog 2 dy dre to y cot x = y sina dy dre cota = sinz. [ dividing by y} TA 42 Put I w y? > 2 33 de la du an dy ase I du with 2 da - - dv + v. cotx = 2 dhe sina dv dis 2 cotx. v 2 sina (-2cot adr Integrating factor e 2 log sina - Sin 2x .. solution: ce 2 sinn I sinne 2 cotx. du Sinage are I V = sinage six du + V. 2 sina 11 Ta Sinne dr 2 sina sim I man (sina) a ban (visine) s de Esine) -Sa sinad N 2 sina da tc + 2 cosa tc > sinzse

6 2 sinuse osa + c sinza TA y - 2xsinze cosat e. a sinze (= v= *) y (%) =) i-e y = {1, if a= I 골 1 = 2.1. sin q cos? + * ca sinh 1 C. I 2 c de : solution: y = 2a sinza cosat 22 sinza