Question

Use the Frobenius Method to solve xy''+(1-x)y'+y=0.

Answer 1

Given differential equation is Ocy" + 1–2)y' + y - 0 - 0 Ilearly is regular singular point of giver differential ean y(x) = [ an x ; Goto be solution of 0 Ꭴ Let mtr nco then n+2-1 so [ (th) an a no mund and y Ź (+2) fn+ 2-1) an za no Substituting these values in 0, ce get E fhea -1) +1) Op 2002- (n+r) an x n+1-1 + (n=2) nth (M+2) Ana no 720 co nti of [ an x 720 Put malam mtn and then finally replacing mbyn, en ) [ (0+2)(n+8+1) Anti x E Mart) anti anth nth M=-1 narl co nth Ľ nt. r) an at t an anth no ก-บ nts = (2-1) ir ao aky + r ao x xr-l (1+2)fm+8+1) Omix + M no * f+2+1) One, ze nth Enth) an 27th - ntr Ana" no 0 no no

L-1 Ao x + [ ((n + 2 + 1) * Anth (0+2-1) an mtr 2 mo By compering different power op 2 m both sides, we get & so (as lo to) which is called an indicial equation and its moto are L-0,0 and reccureence relation be (1+2+1)* Anti (172-1) An = o ; N20 (M+4-1) in 20 (1+2+1)* -) anti an When &zo, then na1 an 20 Anti (M+1 al ao Whenn-o; a 0 When nali Qg 0 aa 1) When n=2; Az es 2 . When n= 3 ) au az 70 16 5 An xnt2 y (x) [aman Qot , X+ Qgx't π no no ao - lox A.(1-x) is one solution of a

Now another Linearly in clependent solution of 6 is given by; Since , Anti an intr- (1+2+1 i 720 d), -) When noo; a, a (1+1)* When mali aa = a, (2+2) When n=2, Az = Og and {+! (2+ 3)2 so on.. y ao x² + a, x^+ + axeta + 9x2+3 ay 0,24 + (2-1) (201) 2+1 (1+0) ght2 x + + (272)"

=) ay Qs arIn (x) + (x+1)*((1-1) zelo" ln(x) + f2t)) - (2-1) 42+) 2 (24) or (2+134 + When i=0 e dy di Ao ln(x) + (-1) a ln(21) + a + x2 + - A. In 1) a ln(x) + 3x + Ao ln(x) + (3-Ln(x))x + + y(x) = be another soll of O. (3-2n««)) x + Ao ln(2) + /3-

Answer 2

X^2y"+3xy'+5y