Consider the eigenvalue problem y" (x) + Xy(x) = 0, y(-1) = y(1), (-1) = y/(1) on the domain -1

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Consider The Eigenvalue Problem Y X Xy X 0 Y 1 Y 1 1 Y 1 On The Domain 1 R 1 A 3 Points Sho 1 (22.54 KiB) Viewed 30 times

Consider the eigenvalue problem y" (x) + Xy(x) = 0, y(-1) = y(1), (-1) = y/(1) on the domain -1 <r<1. (a) (3 points) Show that there are no negative eigenvalues , <0. (b) (3 points) Show that I = 0 is an eigenvalue with eigenfunction y=1 (or, more generally, y = const + 0). (c) (4 points) Show that the positive eigenvalues are 1 = 7²n2 and that cos(n7x) and sin(n7x) are both corresponding eigenfunctions (so the corresponding eigenspaces are 2-dimensional!) for each n = 1,2,....