Question

Consider the differential equation (1-x²)y" - 5xy' - 3 y = 0 1. Find its general solution y = Xar, x" in the form y = doy1(x) + anyz(x), where yı(x) and y2(x) are power series 2. What is the radius of convergence for the series yı(x) and y(x)?

Answer 1

Date Daga Do given differential equation (i- de 2) y" - sxy' - 3y = 0 let y = Eanxh y(x) = Enana y'(x) = n(n-1) an xn-2 ho hi n=2 dift equation So pluging into the (1-x2) Enen-) an xn-2 n- 5x. E hana n=2 - 3 & anxn no - n(n-1) anx ह nen-1) and n=2 in 2 -Ě 5n anxh 3 u anah = 0 ne) → - En(n-1) and & E (+2)(n+1) au +22 h=2 neo n E inanze - 33 ans nad ar Eln(n-1) an +(2+2) Cht) an + 2-5h an- 3an) n=2 A + 292 + 6 9 32 - 59,8 - 390-39,x=0 292- 390 = 0 92 = 3 390 ero 693 - 89, =0 - 93 Tai

000 Page -34 5 han -n(ny) an +n+2) Chti) au +2 m-n-sn-3) + (n+2) (h+I) 2 -30 > anta (n+3) Cht) Fo+2) (+1) h+2 so IN प m:2 5_92 40 au प 453 93 - १२ 콩시동보를 영 पंप है 27 3.lx E (antispana) (2ha) so y = TY ५० M +91 2 M (20+12/27-12-3) राना pe 3) n: 25 मात्र)En+1) 25 -1) (20-3) 2hh Radius of Convergence 20+ h+) (2011) (2h -2) ht 21 Qimen+3)2+)... - 3.1 xn

lim x2 2 (n+1) Radius of Convergence 4 12) = E 2 int! Gn+1) cant) 3.1 2nt3 2n + 3 L 121 Radius of Cony lim ht' an+2) (anti) (2n) . . 3.1 x 2n+3) (2011)..3.1 x (2 Lb+1) lim & (+2) 12 / 2 Radius of Convergent e o