Question

1. (5 points) Solve the following eigenvalue problem, i.e. find all eigenvalues and eigenfunctions of the problem y" + (1 - 5)y=0, 0<<<1, 7(0) = y(1) = 0.

Answer 1

Anso hiven, y" + + (8-5) y = 0 O LA LL y'(o)=y (4)=0 LO 20 Case - Il let a-5=0 Id=5 then, ① turns into Y" = yl=4 y = 44th y (+=0 o= ath y' 201 = 0 9 =o trivial solution an I Ly=0

Case -11 let 3(1) = озо 2 4e"+" е к (ен е*+0) 2-S Zo als obrang has 2 – 5 — — к" where 50 then, " — ня 20 з (3-- к*) + = 0 Auxilary equation & w. No Э м. н- m = ²M Then solution of homogenuous equation is equal to complementry function, -Их 8 = + tо е ' - нечи не-ки : 810) = о з о= (4- Jка 4 = l (H+) о 0 = (eh4e") че KY B) Gemtqeu 2) Э Чso | = 0 trivial solution

Cose - - a-syo ays let 2-5 = M² where MYO then, y" + M² y =0 (0²+ M) y=0 auxilary equation 3 m²+ H²=0 mat Mi Then solution, y a Cos (Mx4 G Sin (MR) yl=-MG Sin (r + MG los (MX) " y 101=0 0= MG z Tc2=0] (4+0) a y = a los (MX) * Yht) =O 3 o= G los (M) If G=0 then y=0 trivial solution gf 470 then, Cosly=0 M = (2h+t) I 25.5 = M = (2n++)?T ។ eigenvalues noteikumi olan 5 + (2n+1) 772 4 and eigenfunction, Yn = “ Cos (2+2) 1 x)