Question

Find the exponents at the singular point 850, cand the fast thire nonbero terms, to the larger root of the indicial equation Xy"-2y =0

Answer 1

* Given that : xyl-2y = 0 Which is of the form: PCao y"+8(aoy't Rear y=0 where Pra) - a, 20 = 0, R(225-2 row lim (x-xo) (20) P （༥） ayxo lim xo x 0 ol = 0 [= Po] lim xyxo (x-xo)? R(a) PC) lim x? - 2 = -2(= 90) 270 so. do o is a regular singular Point, row Po=0,90=-2, the corresponding indical equation is : F(P) - r(n-1) + Port 90 PCri) toin 2 -p-2 - 2 with root 21,2 = -2, 2 D the first solution ý given by! y, ear=131) "[ 1+ Ê an crian] (x) ° [ + air an (2):21] Its co-efficient apart from ao(o)=1, in this case: rani

2! so, n² ancost 27 o ancos ak(o) [KPn-r naktan-x] =0 where M& VNEN Pnxn = x O gau pca 22 nzo ole - O (2 anan = x² Real x? 2 peño 22 n=o a 21 Simplify (D and 2) we have: Pozo, Pn=0 VNEN and 917-2, an=oxne no l{2} factoring out anco) we have n-L anco- ax(o)(KPA-K'tan-k) K-0 n2 INEN

How 0 (0) - Q66) (64 - 3 : C- 2 ) 1 - 2 A A 2 (0) - T Goto) 9, + 4 (0) ( R + 4, ) 2 2 -[1) - 0 4 2 (-2) - 4 Azco) = C Ro(0) 92 + (0) CP, -1.) + 4 (0)C2P, 490 3 [ 4) 20 4(2) - (0) + :: ( 2 ) -- - (6)