Question

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6 -1 S (s – 13)e-48) } = (a) u(t – 4) [3e +2(t-4) – 2e3(t–4] e l s2 - 5-6 (b) uſt - 4) (3e2(t-1) – 2e-3(t-1)] (c) uſt – 4) [2e3(-4) – 3e-2(t-1)] (d) uſt - 4) 2e-3/t-4) – 3e2(t-1)] (e) None of these 7. The general solution of the differential equation 4.ray" + 8xy' + y = 0, valid for 170, is given by (a) C1|2|-+ +c2|21-1 (b) |2|+(c + ca In|xD) (c) C1|2|–1+23 + c2||-1-2/3 (d) |x|--[ cos(2/31n :-) + c2 sin(2/31n|:00)] () None of these 8. For the equation ry"+y+ 12y = 0, (a) Both 0 and 1 are regular singular points. (b) Both 0 and 1 are irregular singular points. (c) O is regular singular and 1 is irregular singular. (d) 0 is irregular singular and 1 is regular singular. (e) None of the above. 9. One (nonzero) solution yı of the equation r’y' + (x2 + 3.0)y' +y = 0 has the form (a) Ženz" (b) Žam?**? (c) Ženz"+ @ Şam*** (None of these n=0 NEO no

Answer 1

If Lif 1414 = fis) then 22 flut-a), ult-a)= eas Fis) fi4a).clx-a)= Z'yes from So ille have to solve 'l (3-13) ļ and 8²-5-6 then using s-13 32 35+25-1 (3-3) (St2) 7 Teos + = Hist2) + 6(-3) S-3 (8-3) (842) =) 8-13 = A 18+27+ B(5-3) put s=-2 -15= 0 + Bl-S) 2 B²3 but s23 -10² SA tO & A2-2 So 8-13 13 ore a 36 por 2 1859 13 stase e at * Fly 384 244 44 s using or = f(x-4)u(1-4)

[ { (5-13) e 4 8².5-6 - 7 ult-4) 3 e 214-4) 2 314 0 4x²4" + 8uyty=0 Pesut y" + 2xy + 220 - 2 If xy" + axy't by zo en bey Cauchay Euler reqabur its reduce to y"(z) +1 2+1),y(2) zbyte)zo whee *ze?] so reduce to Y" (21 47' (2) + 9 20 It's auxillary equation m² + m + 1, 20 m2 +2.764+ 'zo z) (m+)%20 m2 12 12 / 2 keez) Zz Jiz) = (Cat (az) et -:xze? a) zalogens [the log in ] you eat Cat Callage] yura e la mia [ct callegur yeni= 14² [arce layer] yma 127 12 [cat ca in 108

W 1-) 12 10 XBO XI 2 W ! X2 © "у" ( yo 3D n-o) li h a limite mot erist pruiken. So xo is an ovreguber singular farint. x! lim (x-1) limit evind fistely din (x-v* met meer toa 1 limot erat finitely. So x= 1 is an regular singular forint. x?y*+ (n°4 37 14'ty 20 B as x=0 is on singular point of giver differential an. ding (-0) R134 photos tage le forum ) washing L'ofitel Rabe in 24+3 = 3 limit erent fintely, in 1-0 - - in the 12 = 1 limit exist finitely. Sonzo is an regelar sergeber point of diffentical equation het y = 8 On (x-on is a solution of 6 . y = { an (nam) y min-1 y = { an (nem) (mtn-1) - Na 20 n20 maman, nim-2

put all there in (3. Ean (mto) ( m+ n-u) xhtmn + Ean (m+n) xe*** +37an (men) samty + Ean whom o put n=0 and table coefficient of lavest term of n. ie. Coefficent of an a min-1) smas tao ao 20 [mim-1) +3m iJs0 - 20 to 2 m²m +3m +120 - m² +2m+ 120 (mt) 220 m=-1,-) Roots of indical een 6 are meant ; m22-1 so by y ; one non-zero sebutar of you Lan (x- 0)***