Question

(a) - Find the areas of the surfaces generated by revolving the curves below about the x axis: (i) y = 13/9, 0<r<2 (ii) y = VT, srs (b) Solve the following initial value problems: (i) z’y' + 2xy = Inz, y(1) = 2 (ii) tone = 2 + 3u, t > 0, u(2) = 4 (iii) ry' = y + x2 sint, y(t) = 0

Answer 1

3 건 در حال a) i) Given y = ,야라스2 2 NOW y 3개 건 3 0 da The surface area 4 solid of revolution about the x-axis is given by 고 s=5 2TY X 4 3 et da 건 2. - + x 3 = 0 3. = tr ) 복 da + +1 W.K.T, S [f coxyf'cm) de [fxy] nt 요 + 2 1 +/ 2 2 4,312 ㅠ 2. 3 1+ . 1 ( 0 312 IC (1 ㅠ 3 ) [(t 3 [1 - 1] (15) = () ㅠ 3 78॥ SI

3 X i) Given =원 스건스 16 + Now 서 a da solid f revolution The surface area of about the X -ars 8 given dx ss +) 개 5 1+ 뿌 dx xto 이 13 X FT =ㅠ 2 | +xt) da ㅉ tele tolai ele 11 ㅠ (++)) ) 와 HE 9 () e (나) ] TI 품 ( 411 (c) - ( r) - (56) 28T

lux , - 2 ow zl. y - x2 lna x² 2 lna s 17 h dx 2 sub. 'c' value iw (1), we get b) :) x²y' + 2xy 7(0) Dividing with a bos, we get 읖 lina at is linear D.E iw 'y': Herre, p(x) = 2 e & Q(x) = enx² i's Ii Fae x The Q(X) (I-F) dx + c lna 4 x =7 (2x) dx tc s lux date (lnx-1) + Ca(1) put x=1 & 2 = 2 in (1), we ge get 0-1 + c => c=3 7 (I.F) = S ا X => 4x² = Given 3 (1) = 2 2x1 = I (boce) -1) + c = 2 yx = x (lnx-1) 4 3

ii ) Given t du dt tt zu, u (2) = 4 => t du 3 u = +2 dt get Dividing with 't' on bis, we du 3 t u=t dt It is linear D. E in 'u'. 3 Here p(t) = { ¢ Q(t)=t dt -3 lnt 3 5-31 In = e as I. Fa 3 = 3 t - The general solution is given by u (I-F) = SQ (t) (I.F)dt to sy u(t) = st. 1 lt + c dttc E ²+ 2 -2+1 tc - 2+1 u - £tc tc E 317 3 t 3 2 u = => - t t ct + cty (1) Given u (2) = 4 put t=2&u=4in (1), we get 4 - 4 + 8c 7 8 = 8C => C=1 sub. 'C' value in (1), we get u= -t + tu 3

iii) Giver xy' = gt ta'sinx, y(t)=0 Dividing with 'x' on b.s, we get g' 1 / 4 y = linear D. E in 'y! it & Sina is & 8(x)=x sina Here, pea) = :: If empecas da Set) de laz love' e' = clona = * x The general solution is given by 7 (1.F) = ſq() (1.F) dutc =Y () =S Asina (%) dat C casat => Y Giver 7 (T) = 0 x = T & y = 0 EW (1), y a . acosat Cx-7 (1) he 2 get put => C=-1 - TT COST + TTC -T (1) + C => TC = -T sub. 'c' value in (!), we get y = - rlcosx-xl