Question

Question 6 Consider the second order non-constant coefficient differential equation y"- 2ryy= 0, (4) and a power series solution with the general form y()=C n-0 relationship for a,. 6(a) Find a recurrence solution 6(b) Find two linearly independent solutions of (4). Show that is a polynomial and obtain the first three non-zero terms in the series expansion of the other one

Im having trouble understanding 6(a) and 6(b)

Answer 1

+ 4-y-o CD Let 2 n anxn-} n o n (n) ann-2- 5 n(n-i)a, ^-2 n 2 Co CD n (n-) anx?^ S 2nga an xr + E 4ann=o CD 2n anzn 5 4qnz nEo O = 2nt2)(n+1) an+2 Y - n o 2a20 t 4 G (n+) Ln+2) an+2 xn+2 (2-n)an t x1-o 92 2ac=0 2 (n-2) an An t2 = 2ao b) O q2 =) a4-0. (2n ) (2n+1) Ten-3) azn- m (2nt) 2n- 2ao22 n= C 2n-1 ao(-22 t 2 92n-1 x2n-

ut ,)-23 , C) Dotynomial 2x2 2n-) ut nニ) a3 11 a3 ai as 30 ai a s 3 As 7 3.7) 2n- a2n (x7 lzu ai. 7 21 o 3 o 7 3 y2) S12 2lo 30 (A C, CI-22)+ C 2 7 - X 210 3 O wlx