Question

show that the equation xy"+y'-y=0 has a regular singular point at x=0, find the indicial equation and its roots how many independent solutions does the equation have ?

Answer 1

Coven differential equation is ay" + y - y=0 y - 1 1 2 =) Y"+' 20 2 Comparing it with y" + P(x) y't Q(x) y D - P(x) = 1 Q130) = 2 2 2 Ble 3 P(x) = ), ged. 8(x ) -X clearly, xao io regular singular point of 9 Now, Let Yx) = solution of 6 E am nth an a be with Ao to no ntrol then y Ĉ (+2) as a Y72p nth-2 y" [ {n+r) En +2-1) an x and 720 we get nth Su bati turting these values in 0, Ć (+2) 6n +4-1) an 2 nth-1 ann' ob t Ź ) on 242-1 no no mao m n = m+) Put Mala ean implies and then replacing mbyn,

< (+2+1) (m+2) Amt, a w Onant N =-1 al r-l nir =) 2 (122-1) ao x t i ao x + + Ć (0+2+0) (0+2) ant, ac nao E 1+2+1) Cat, 2002. mo Ananth = 0 no no nur X an o W 12+15° ant! (1.*-*+2) do 22-1 t (62493 no wege 20 Now by equating different different power of a on both sides (m +8+1)* Anto - an and Indicial equation be 2920 08 Go to ) co ( 28 moto are 2=0,0 and differential equation 0 has two Linecely independent solution