Question

Consider the following boundary value problem, y" +(+5) y = 0, y'() = 0, y(9) = 0 (a) Find the eigenvalues. (b) Find the eigenfunctions. Take the arbitrary constant (either cu or c) from the general solution to be 1.

Answer 1

with :yllo)=0 Sl: Given Y"+ (x+s) y =o. y (9)=0. The given Probe boundary verlue Problem is Sturm-Liouville Problem. 9 Case Ii let X=0 Then became y" asyo sowing we get Y(x) = C, costs x + Casinssa. - y'(x)=-4,55 sinssx + GTS costsx- a C2=0 Now y(a)=0 y (9)=0 a Losto (9) at q sinfs (9) = 0 C Cos Ss (9) to [Q=0) = Ciro ů C2=0 i ③ became d=0 ů There is no eigen function corresponding to =0

(i, et is -ve). Case II: we get y " (solving me get (ts) y Ju²5x - Su ²5 x = y(x) с с е y' lxl= clues 425 x ( [u's - JM²5k Now y'(o)=0 » c, juzs C₂5u²s 20 Circ=0 (1=2 & y(a)=0 I c 5u²5 Tuzs - Ju²5 qJubs -1u²5 - Ce - Се [:c-4] & Ju'sto 0. = clarus las CO > C=0 =) C = (2=0 (:,= (2] - 5 no }(x) & again here da,-u? eigen function corresponding to

we get -- CO on Case Il if x= u lie & is the (uus)y soluling we get y (x)= G cos [u²t5 x ( sin Ju²5 n. - ③ g'(x)=-6, 842+5 sin sults a tu su²ts cos Jutsa, + Now y'lol=0 С. = y (9)=0 » Ci Cos 5u²+5x9=0 1:4-5) Then uk will again get Now if & Cizo no eigen function G=0. so we Take cito = q los 5u²+5=0 Cos JM²+5 =ol - (2n-1) : 2 =) 5u²+5 u't s= {(2n-1)) u? = (2n-1} 2 5. 4 ♡ be came 5

2. Y(x) = C, Cos ((22-1)) x (as we have to (i=1 * ] Hake g(x) = cos((2n-1) [f] & t="²= (2n-1) 112.5. 4 Hence In(n) The required. eigen function with corresponding eigen Values An are given by Yn (x) cos ((21-1))* An= (2n-1) Il y us. n0,1,2,3,-- Thank you