Question

Please answer all parts of the question and clearly label them. Thanks in advance for all the help.

5. An eigenvalue problem: (a) Obtain the eigenvalues, In, and eigenfunctions, Yn(x), for the eigenvalue problem: y" +1²y = 0 '(0) = 0 and y'(1) = 0. (5) Hint: This equation is similar to the cases considered in lecture except that the boundary conditions are different. Notice how each eigenvalue corresponds to one eigenfunction. In your solution, first consider 12 = 0, then 12 < 0, and in each case show that there is only the trivial solution. Then, consider 12 > 0 and find the non-trivial solutions. (b) Prove the orthogonality property of the eigenfunctions and obtain Soy (2) dr. In other words, for the eigenfunctions yn() in (a) you should show 0 Yn (*)Ym(2) dx = T for m #n constant (you must calculate) for m=n JO (c) A Fourier expansion theorem: An arbitrary function can be written as f(x) = 2oo anyn (I) on the interval 0 S 5 1. Using the orthogonality property in (b), develop a formula for the an.

Answer 1

k=nT - d² = K²=n²7² an= ntt , NEIN. Hence eigen values are nit" DEN . Eigen function on = n o ces (nl) Nett fe for 20 yoxst [(b) of you y (x) dx = ces (nix) cas (min) dx. am) tet Tix-F Tdx = dt وم ""= به محك Tosent) toy (mt) ott. Tces ontmtf.ch - gT [ sin intm) t- L htm - sin(n-m) t nom + u - f Yon bosques des af forcote=- lit niem then it yn du = f ces (RITA) da itointi DET VFCscent)] dr.

Tot + Sin2mt) ㅠ 4n ]。 | | 프+0 - = = insticission hen mm 1'이게 일

. ( fix) = { an Iri (x) as an as (?IT) no Inco Emot zmiany (كلاكها . I fourier ? z expansion 57 . . f(x). Cay (21x) dx NA GUESS