Question

Both power series solutions of

y'' + ln(x + 1)y' + y = 0

centered at the ordinary point

x = 0

are guaranteed to converge for all x in which one of the following intervals?

(−1, ∞)

−

1

2

,

1

2

     (−∞, ∞) [−1, 1]

Answer 1

į an (n) uhy Sely Given differential ean y" + In pell) y + y = 0 Power Series & (2) hro y = { anth Now differentiate egn (2) worn to be Y' Again differentiate egn (2) wonit. lll Ean (n)(n-1) 21h-2 (2) h=2 Now putting egn (2) , (3) 4 (4) in eqh (1) then get + an (n) (n-1) 22-2 T + ln (3-1) ( 3 an (oh) th ) (one place nant) 1962-1) 334) Ja (14) (n+1)x) E anth (Replace n=o n=ht2) (anta (hta) (n +2 -1) de anti ho n=0 4h no

Ant 2 (2+2) (n + 1) + In (2+1) anti (ht) + an/Hh In ]wetre T neo anez (h+2)(n+1) + ln (4+1) anti (n+1) + an ao - Ant 2 (h+2) (n+1) an - In (4+1) anti (nl) =) anta = -ana In (4+1) anti (n+1) (h+2) (htl) diuided beth side by Chez (n+1)) -an - In (041) and (hal) (h-42) Cht) an42 (H=0 given ) L anta (5) (h+2) (hti) ao Put nao ag ag de - -au 03 Put nal -9, 31 1 put n=2, aa 99 H(-4) ► 04 = + a = +90 (4) (3) 24 & 9o 41 laring value of J ac 2

Put na 3 as 93 (5) (4) - - az A5 11 = as 190 1 As 51 bring power Series 401) 8 andh 17 Qoto + Q, H & Agt? + Q 43 +99414 + agus + heo ૫ (ઝ ) ao tail & Ag + Qz H3 + A4 H4 + Agust- using nalues of as, a , A , Az, 9A & as in eah (6) Y(H) = ao tan + (14) *? + (194) 18+ (04** + (G HS 41

Y(1) = ao H 21 ? + *** + 7-abu ta, H-43 3 Ao Cubot a Sincen 7 4 () = ao Car (o) tai Sin (o) ( Hao given) Y. ao (1) + a, (o) 96) = 4. (ay) % = ao constant Series - 4. T Series convergent And we everwhere know that constant simply say that say that the given differential egn y + In (441) 4' + y power series convergent in (-co, ao) = R We using