Question

4. (a) Find and write down the general solution of the ODE 2y" – xºy=0 in the form of a power series about x = 0. Only include the first three non-zero terms in each of the two linearly independent solutions (in an interval I centered at x = 0) that you obtain. (b) Check that each of the two linearly independent solutions you found in 12 part (a) individually satisfies the ODE, up through terms of order x'

Answer 1

Sol. 4. a gy" - x3y=0 y = Eanan be its power series solution. 6t n=0 3 then x y=o o Qy" 2 § n(n-1) ana" n-g x {anan nco n=g n-a nt3 E ana E an(n-1) anx T nco n=2 n+3=0 ท-4 2n (n-1) and ana T MEO (nen-3) 9 ( nonta) E 2 (n+d)(14) Antgen - 2 an-д х 2 na3 nro 4ag + 12&3X + 24040' + (949) (941) Antg - An-z] sx"=0 n=3 24 au=0 다. 492-0, 1203=0, 2 (n+)(n+1) ante l ana=0 Vn73 A2= A=ay=0 d. antaa an-3 2 (n+1)(n+2) TI

then as - ао 11 Go 40 2.4.5 06 ai 11 60 2.5.6 (2:02:o) a7= Az = o d. 6-7 aga E o Qq=o ao !! do = As 2.9.10 ας 180 7200 at ai au 11 !1 a6 2.10.11 220 13200 9125 at 0. 6 al "+-- 10 a ас t Rand" ao xstaiget 40 60 138 0 Tot air + ya ㅠ 7200 nzo =aol of ai 5 1+1 at 1 xt 40 720 +--) x + x + x bo 13200 2 Yg y, 6 न 5 Now Yi 5 Itx 4o 16 2 7200 Yg 60 13.400 .10 4 १ + I y y = x gy + To 1200 8 720 3 여 + x + Y;" = 8 X 80 You el 4 9 1 120 2 2 3 8 3 13 X gy," - x?y,= 3 x + x X 8 a 40 -- 7200 T ५० 7200 term) (: each left term is concel sy noxt higher degree 14 14 a xtr x3 Ya ² - u 21 gy" 60 60 13200 1320