Question

Given that x =0 is a regular singular point of the given differential equation, show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about x = 0. Form the general solution on (0, ∞)

2xy''-y'+y=0

Answer 1

Given that x-0 is a regular singular point of a the given differential equation, 2 xyl-y'+y=0 The given x=0 is a then above DE is sove by frobenius method Let y(x)=x05 anx= regular singular point ananto is n0 no equations solution of Then satisfied équation 2% E an(n+8)(n+8 -1) x n+/-2 0+2-1 - Ean(n+8)x NO no nty + anazo n-0 TO 22000+0) (n+/-1) senty- -Ian(n+8) antola n=0 + Eamento => Ž zan (604x5?- (NE) - (ntoj] xn+s4, a no n=0 anx x ntr Ź anboto) – 4 (0+0) 4n+8-1 + I anx nto so no

2 equation of Now find the indicial for equating the coefficent least degree we have a[282-48] = 0 goto 2018-2)=0 8=0,02 270 finally 8=0, 8=2 80,2