Question

x?y" + Oxy' + 9xy' + (16 + 2)y = 0, y(1) = 0, yle7/9) = 0

(a)	Find the eigenvalues.

as a symbolic function of n

(b)	Find the eigenfunctions. Take the arbitrary constant (either c1 or c2) from the general solution to be 1.

as a symbolic function of x,n

Answer 1

Given equation is x²y"+any+ (16+) y =0 Let x-et at is the new independent variable. then xy" = D(D-1) Y D=d JE axy 30, © becomes aby D(0-1) y + qoy + (16+) y = 0 → 02-D+90 +16+ +16+1) y = = 0+)0 mt - Ae Let y = (70) be the trial solution of . Then corresponding auxiliary equation of ② is (m + 4)² + x=0 n = – 4 ti Case !: ico then the solution of is (4+Fxt (-4-Ft y = Ge 64+1=1) (-4-5-1) 34 ) = 4 x + a tore (x=et)

O= 4+ g = G = -2 Now, y(1)=0 gives y (eda)=0 (4+5 gives 0 = ce 9 (+4-FT/ the (4-5-x) Na 7 C X7o, 64+50W 645) los te 4 = 0 po ce 3 so, y=0, which is a trivial solution. So, there is no eigen value for a co. Case 2: x=0 then the solution of is -4t -ht t y = qe the = y) = aut ce ln xt – @ Casaret) Now, y(1) =0 gives O=9 - Ana y cena)=o gives of C 7 e C2 = trivial solution. So, & becomes y=0, which is so, there is no eigen value for x=0.

4t case 3: then the solection of ② is y = 4 sin (JFX t) + Co Cos (Stit) é y@ = G Sin (Nar Inces) + (n_C06 (J73 In ()} & 1 Now, y1=0 gives .-4774 ye79) =0 gives 7 G Sin T+)=0 So, for non trivial solution, a to o – than no sin ( o=4 so, sin cha JFA) = - Sim (na) I Jtx = ил T n=1,2,3,... 5 1 = (an), n=1,2,3,... an = 963, n =1,2,3,.. So, the eigen values are the eigen functions f (x) =cn sin (anln (*) are n=1,2,3,.. 2 le, J. () = Cn sin (an Inse) , n=1,2,3,...