Question

2. Determine the normalized eigenfunctions of the given problem y" + ly = 0, y(0) = 0, y'(1)=0)

Answer 1

g" + xy =0 -0 cose Į when A=0 - then equation 0 becomes = "=0 2) Y = AXTB / Where A and B care © constants ) " y(0) =0 al B=0 = y = AM =) Y'ZA "yl1 = 0 =) y'= A =) A=0 =) Ty=0 =) There are no eigen functions corresponding eigen value zeno. Case II when X = 12 then equation 0 becomes y"+12y =o The Auxillary equation of ① m2 +12=0 = m = tim The general solution of o

y c coolu u) + d Sindex) - © " (0)=0 => 0 = CX1 +0C70 Hence, the equation ® becomes ya d sinler - diff" equation ® woro to 's' y = doll copex yl(1)=0 = o- du coole a 4 cool =0 ordzo coollo . Cool= cop (21-1 ja - M = (2n-1) a Hence and eigen functions . 2. 2 IT the eigen value You do sin ( 20. Du Monde

Case to when X=-112 Then equation 0 becomes - y"_42y zo The Auxillary equation of o m2_42=0 ) m= Il

+ y = E eur +F é un o y cos=o -) 0 = E + F — © =) E = -F. Diffneqn ② wor.do 'n y = Elleen_Fue un as y'll=0 - р - Енеч - Ене- и » Eletllte Méll - Eureure-lzo = E=0 3) F-0 2) 1 y=0 2) There is no e i gen fonction corresponding | X=-42