Question

Problem VI (15 points) Find the eigenvalue and eigenfunctions of the following boundary value problem 2" +44.x = 0, x'(0) = 0 :'(1) = 0.1

Answer 1

21" +420 20 n10) 20, 2(1) = 0, Case I Let azo. Then n(t) = At Att B. now allElza x100=o a Azo, . i alte B. @ : when eigen value o legen function are count Real want!

Care 2? Let aco, and a=-p2 PERS. :,4+420 20 an"-40"n=o. the soen is atlaçet e 2pt. 1. Then rills 22PG e2pt_apgerept. x"101=0 ) arazo. 2e11220 7 622P_Gé 2420.. Thor 1 Then G=6=0. i, (H = 0. Hence a cannot be less that zero. case 3: Let a>0 anos Then the solution of "thanzo is X14) = Gosarat +Ca sinarit Then riles = -250 G Sinzat +257 G GS 25t Thon n'la)=0 =) 25*C2 20 =) ( 20. peftty 2 08201 x) = -2 G Sin (2576) How x61320 =) -250 G Sin (250) 20. 1. If G =0 then tant ale) e o Than let ato. then sin(2500)=0. 2fann nez. Tahun : Eigen values are man . an- north former Eigen tametions are Cos(2.0.1 t) Eigen washine &20, Eigen = cos(matt) tunelim - B, BER. Eigen value an=nr for mer. Eigen functions are Cos (mat) for nez.