Question

Solve the following IVP 4y = y = xex/2, y(0) = 1, y'(0) = 0

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Abdul-Rahim Taysir

Answer 1

x/2 472_120 e,,C, care Condant Now, for non-homogeneous Since. x/2 general soin of , we - each element and thes, the A, B are Ay"-y = né The homogeneous part is, ay" y=0 Characteristic polynomial for Dris, yp= # general soln for 6 rio, for 8 ris, Jn (x) = get + ezetz arbitrary part, consider the Undert undetermined coefficient set si={rets, et is also per present in multiply of si by a S*= {2+exs, retty particular sol" t's of the form, confants. Yp = Ax² 124 B xe/ Yo's y = A x264/2 (4A+B) *]xe A] (24 & Putting you and and y Ax²,"l2 + (8A+B) x 22/2 + (8A +848) e*2. x 12 A x² + (8A +45) in both sides cue have, Equating like power x2 Be OTO Ах? h. ( + 24 Dee ya + 2 2/2 개그 2A 2B + + B + 4A 2 4 2 4 2.42 A 4 X + 2A + B 1/2 хе + e 4 1x [B + 2A] 242 ginen equin we have, in A x² */ By el 2) = xe 7 8Axel2 ex/2 a/ ze terms SA 19 A = - 3 SA+46 - 0 7 4 B = -1 22 242_ x 8 22212 => B = -1 Yp (*) ". The general sol" for the given diffn equlis, ya) - e, e 2 x ex t + * x², 2/2 - 1 xet/2 y (o)=1 of autcz

Now, g 늦의을, 월 (2422~) 22 .g0)=09 0 = 4 - 국 16 C2 CRAM C. N II 이에 A 40) 키 oa 1. + J - 를 > C: Alw 22 1닐길 I 가 the soln for the given IP , 시고 개2 ga 42 A/W x² é e + 국 e + + 국기와