Question

The differential equation X2 - x) dx +ny = 0) is known as Laguerre's equation. a) Obtain the regular solution in the form y(a)= B, 1+ (=nye mcn = 5125" (n=k+1}x k=1 b) Show that this solution is a polynomial of degree n when n is a nonnegative integer, and verify that the choice B. = 1 leads to the Laguerre polynomial of degree n, with the definition - ... + n! L,(x)=1 – () * + (3) ... +47* represents the binomial coefficient n!'[(n – k)!k!]. Where (*)

Answer 1

-0 -)0 m L let syu+(1-3) 4't nyo (1) · Sinu 2=0 is regular-fingular point af (1) hecaure lim x(116) = 1; lim (A) = 0 Hence we will assume that the seriel suluhun ud of the form you? = I am a rem , your = Ë form) am xatm-1 (3) yooms Ž Cormo comm-edam xilm- 2 60) pult these lealued in (1): 0 = x ê [ {crumɔCorm-12am x tn-2 47 + C1-x) Ž {crim yama'tmis icho chic thŽ {am come 0 = @ [Cytm Xx+1+1 ) +(*m)am sc*tml + { [n-crtm) Jam 27+m Os Cotm)?am 2 tm-1 +Ž (n=r=m+1.cemy zeltm-4 op?a, x*t + [catmam(n-v-m +Dam-,3 ,1m-, · The indicial equahan 720 => m=825550 Henue os Corm)? am tcn-r-m +1am-, with ro = Am = [mt-J Am-1 ;m21 Take .au= Bo Carbitvary) ·0,= -99. = mz @o Q = + n6i+n) ao (1212 .az - n(n) Cn-2) 9 (13)2 1 0 (0) H)( 4)..(h- k+) mu mai m2 Gr= (n (men (Lk 22

• Thus pulting these values in (2) with 820 ; we get your-yice) = B [1 + Željk nen-A2C0-1=CH-ban) ace] • letmeneIn , then Obuwoully, 0 - GMH = Am+z = ....- Amts. 4, (x) Contains power of x dess than or equal to man 7 Yix) is polynomial of almost degree in'. Now take Bo=1 ; then from abou result with Y,= ln Ln(x) = 1-{ņ432? +6012 {mentos para +con = 1- c, >+ 2 + Content 26 - 27 x + 2 y 2 -- ++ C-vom zn. pero