Question

(1 point) Find the indicated coefficients of the power series solution about x = 0 of the differential equation (221)"-ry +y= 0, y(0) = 9, y'(0) =-8 T +0(x°) y= 9-8+

Answer 1

Solution clifferautial eguaHon Ci ven the Ct)yry'+yo, yo)-9, yto)--8. focer le Hhe (4) series represeutation i8 (2. y)=C, +x + 39 x2t 40x3+ 5C5 xt4 6C85 7Ca xt 8Cgx7+ 9c,8 "()= 6X+12 Cx2+30 C 30 Cx 42 x54g6Cgx672 Ca + (4) of equahon 4), )&(2) in egh ) than werget now subshdute the values c22+1)(2+69+12C4 22+205 z3 +.30 C2 xt 42c x54 56 x42gr) x C Ct2Gx4-3gx2+44x3+55x t+ 6Cex547C 6+8C3 x7+9x) Cot Gtx x+Cgx4+Gx 5 C7C3 x)-o (2ct 6gxt12Cx420csx3+304, x44 42ex5456 2x-) + C20t632t12x+20csx34 30x4+42 C 5+56cg x472 C7) x(G+20x+3x2+ 4c4x3 45C, x + 6q 2 72648gx+sc ) + (Cotgxtex2+x3+C4x4+5+ C; x6 +Gx7t cg x8+x--)=o (a26312 C4 xt+20 Cx 30 420+56 cg x8) C2g+ 63+12 C4+206x34 30 Xt+ 42 C x54 56 C 64-) - (Gxt29x2t3g x2 4xt5es 5+ cQ z4 7cp xt 8328.) +c Cot Cxt ex2 +3x3+x C524 C6x+C7 x+ Cgx8 )- o (2c+Co)+x(63-C, +G) +x- (22 +12 04-22 +)+x3(603 +20 cs 3c3 +C3) +x4(12 C4-t 30 C6- 4C4 +C) +x(20 C5+ 42 C-5ste) 2(30 C +56 Cg-6+Ce)+ equaling get on both sides Hen we fticiouts Hee Core nou Co (5) o= 02+56 C3 0 6C3= 03 C4= (6) () 9A 12 C4 +C 0 12 0C5t 13-0 C553 g Cat 30cg 0 Ce= 42 C7 +16C5-o C =- (8) 3 C4 (9) 2C4 = 30 16 CE 42 (l0)

and 30 C-5C6+56 Cg = 0 56Cg+250-0 -25 C - 1) 56 given gto 9 and x-0 in egh ) Hen we get he initial coudidion. Sin ce yo-8 put gio)- 9 =Co Co-9 mQu & put x-0 in egh(3) Hhen uwe get gto)-8 = G) C = -8 hen ce, X T 12 1 -xo=0 S C5= 9 3 X 3 CE = - 10 C = 45 45 t3)- 25 x C8 = 896 g0 56x16 56 Cg. In eghlo) the values hou subs got hen uwe +3 x4 0%x5-9x6+oxt+45 80 9 t 0x3 896 yr)9-8x 45 896 4 g0 + 9-8x-