Question

Question 4 A power series solution about x = 0 of the differential equation dy dx +(x+2)y = is Select the correct answer. y=00(1-2x+2x2-3x2) OnY=60(1+2x+3x2+3x... Oc. None y=a01 - 21x2+x- Od

Answer 1

The dæ Let n=0 given equation is done +(x+2)y +(x+2)y = 0 y=[anan be the power series solution of the given equation about x=0. : dy da n-1 .nl Inanan - I nanx' n=0 n=1 " By 0 n-1 § nanze + (x+2) anan =0 n=0 n=1 (n +1) Anty 22" + ananti +2 anze" =0 n=0 n=0 n=0 ant n=1 n=0 2 (nto anti Banyan +2 +2 anden = 0 7 Ž{n+) anxi + Qan-n+ 2an3x2" + (Q, +200) = neo n=1 200 ,, a, + 20 = 0 Ap & (n+1) Anti + An-1 + 2 An = 0 Ź anti +gar CS Scanned with CamScanner

3 - ao 2 = 7 Ag = :: Qq = - † (20 +221) --- (0.-22a.) + a2 = а, } (a, +222) = - $ (-200+2300) § 2o Hence the required solution is y = 26 + 2, Y + 2x +0313+... + y = 0ot 42 a.) x + (39) 22 + (-4) x3+... → y = 20 – 20.8+ Ž 0,25 § 40 x3 + > y = ao (1-2x + 3 x ² 1/4 x ² Hence a. is the right option 3 3 + CS Scanned with CamScanner

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