Question

Q2. Solve xy" + y' + 9xy = 0 in terms of Bessel functions. Q3. Consider the power series S. (x) = 2 0 (1) Find the radius of convergence for S..(x). (ii) If S. (x) is the series expansion for f(x) = S..(x)? Explain. [6] Can we compute f(12) by replacing x by 12 in [4]

Answer 1

ran Given equation is ayl + y + gry zo 6. multiply by :x , we get rid y " t scy' + gxy=0 , this is a Bessel equation of order zero substitute :y= $19,x) = ax? + [ an orth in , weget (in you anxoon. y'= Lan (ranyzanne *y= Lan (2+) (549+1)xeanu) . :: 0 < x ancoran) (541na) z zana) al centro), pano) + 922 ( 42*") -0 - Î Q (84%)&rin n -2 n2 n = rth de na rin o +) lothinth ar (8th) 2 thI2 tl gane to n - 2 hal - não an (r+h) (o+n-1) 2004 " + (a, + + 2 anoth) net . n=2 ganz so orthta herth 7 n=2 g an x Mazdan ) n=0 n=2

n=2 Ő = [ (antramy (8+n+) + «n (run) + 9 an 2) + 9,6132*1 = 7 An (+n] [v+n1) + (y (7+2)+gen.co ffor n=2,3 - 5 21(741) = 0 — = ar fram eran - 5) – 9019 2 = ng ama (+)(1+1) 6+1)(7+-1) +(r+n) = 6+n) (x+n+1+1) =(+n) (r+n)) . An= -9 ana , for n=2,3,... [ 0, (04)) =0 - fring? Now, the roots of indicial equation do-1) +8 = 0 are 0o (28+0=0 820 8=0,0), put 820 we got, a, (0+1)=0 = (0,20] $ An= -9 m. 2. , for n=2 , q= -9 = , 12 = 9= -3450 - 10 nely a Dystore og flere), n=5 4 1 56 * 4 = to ), "-7- ==0 n² n=2,3,.... and so on

then n= am to (odd) then il so, we have, if n=2m (even an = am = (-9) m a. & (for m=1,2,.....) 22m (m) ( a = - 9 ao = (-9) a (since, 1=1) f for m-2/ Ce we have a calculated, a = (-9) ao – Egle a. 226) (21)2 2422 similasly (n-3) a= (-9) 9 12 -9 & (9) 2a. 4 2 22 g3do Z6 62 Igrao , ( 26 ) - 2x2x2x2x2x2 = 2 = 4² x 22 and so on. so on.

m 22m (m12 m=1 onere cm - mio m= o so y = 2 +on+ go" c. m => y= 2. [+ = , Tom, chowe Jok) is the Beassel function of the first kind of order reno. Sool) = Ï Ï egym compare it with i n we get 8=2 we know that com converges for 10141, hence 2009" converges for p c » lk 26 - Radius of convergence is [6] (: If series are converges for lakk then R is radius of converene) led i (aya = fale , just now TO MEO YAO no no Hence now m=o we have seen that this series converges for puk 6 6 < x < 6 & 128 (-6,6) Hence, we can not compute f(12) by replacing x by 12 in S(x).