Question

2. Solve each of these ODEs using power series method expanded around Xo = 0. Find the recurrence relation and use it to find the first FOUR terms in each of the two linearly independent solution. Express your answer in general form where possible (well, it is not always possible). (a) (25 marks) (x2 + 2)y” - xy + 4y = 2x - 1-47 Note: expressa in terms of power series. (b) 2x2y" + 3xy' + (2x - 1) = 0 (25 marks) (i) Show that x = 0 is a regular singular point for (b). (ii) 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.

Answer 1

See dear these all are different lengthy problems. According to HomeworkLib guidelines I have to solve only the first question when multiple questions are given. So I am solving first question. Hope similarly you can solve other questions.Rate it.

a Given (~72) y"-ny't hy = 2x - l-un (0) het y = í anne then n- m2 y = Enana y'a In (n-11 anx' 1-2 nal =) n-2 ny (art2) § nien-1) and n I nana ny na ta n T ana hro 2n -un n-2 Z menur) ansin t e 2n(n-1) and 다. n=2 n=2 ž naisin tłuana an no 1-ya ne n =) Žn (hal) an ant Š 2(n+1) (972) antza n=2. nao a n - za-(1-uaj -Enana te 4 ana .no n=0 n = 2 =) (harty as ) + (12az tsain + [2C0436m +2) antzt (manan+4) an]x? an-(itunt (unst (unst-.-) 4 (aztao) + (129z+zar) t E [zanti) (n +2) Andet (mi2n +4) an xem = =) 4 n=

8u - n=2 -1-24 + { (-unan Jince this in an identity 4 cartao) = -1 =) A =-4 ao -2-3a 12a3t301--2=) a za and 2 (n H) (012) antz +(nr_anth) an 12 4^ 4 for all n712 (rea. recurrence relation) un T Putting ao or antz = -(nr anty) an z (1H) (nt2) ns2 in n=2, 3, 4,5 ។ we get -40₂-16 o)- } ay = 2 x 3 xy 5 -) ay = 8 as= 7 az-64 -2-30 12 2xu85 by 377 =) as + 7 6 LMO = -16 (th (by ) ao 6 - 5 -7.(272a ai T60

ab -12a4 -256 2x5x6 5) - 1/3 (as op) - 64 15 30 u 97 120 -1995 1024 at = 2 x 6x7 19 .377 240 рч 21 2 38597 2016 19 1920 au- 8 no 3 a ( 377 240 5 y=. E aux -as tad taant- given y= Gotain+/-/4-2015+(-/-tan? +(- +90) ry+ + 77 an +(-497 - do mo luft (-238597_1904 xatt ao [l-ar tou [- 2 + 72 1971 - +6+-+-+---) Creaser (ao, as are aret 20160 1920 = ao t ay G 6 R. 30 - V