Question

Consider the solution to the IVP y' - xy = x; y(0) = 2 Find y' (0) Consider the solution to the IVP y' - xy = t; y(0) = 2 Find y" (0)

Answer 1

First of all recall that the differential eq? of the form dy + Ple).y = Q(x)—AH(Where place s9029 da are function of 'se) are called linear differential eq and its solution can found by multiplying integrating factor (2.f) = { poexda solve the given pooblem Now , we are ready to SOL Given =X 1 g'oxy = y (0)=2 > dy dy +(-x) y = x , yo)=2 with above en (A) PCX)= -x and any = x da Comparing ( with

Now Integrating factor (Tºf) = SP(x) dx e S-xdx 11 e - (3) Te IF e multiplying Iof both side of (1), we get - 2² *(* +(234) = me e 2 2 x.e => R. dy -x.e?.y da x² q da = 1 2 ) ily e we get wog.t 'x' _22 zda Integrating above et ax = fxé S a la ly

2 - => y e ES ne e da put -22 all t =)-2x = = feat =- Set do dt वर -Xdx=dt => x dx=-dt sret t to -e? atc -e , where 'c is constant of integration x t.e -x y e :-02 to = 2 => y -e as 2 t x2 e e ५ -17 -1t cea (*) = cearl => y But given y (6)=2 x=0 => y=2 using in (*), we get e.e when 0 = 2 cerl => 2 =C4 >> C=3 Tintini w ........

Using value of 'e' ein (f), we get g= 3 et 3 e²-1 2 ye yox) = 3 eta 2 Differentiating Won't w 를 so y'(x)= 3x y'e7= 3 et 22 -0 = 342 lywo 2 22 + 3x. 2 2 Agaig differentiating y(x), we get y" (oc) = 3 é 4.2x = y(x) = 3 en y"60) = 3 e® +3-0 y"0)=3 + 3x² a e So Hence we get y (o)=3 and gl (o)=0 POLI