Consider the eigenvalue problem y(r) + Ay(x)= 0, y(0) = y(1), y'(0) = y(1). (a) (3...

 

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Consider the eigenvalue problem y"(r) + Ay(x)= 0, y(0) = y(1), y'(0) = y(1). (a) (3 points) Show that there are no negative eigenvalues > < 0. (b) (3 points) Show that A = 0 is an eigenvalue with eigenfunction y = 1 (or, more generally, y = const # 0). (c) (4 points) Show that the positive eigenvalues are λ = 47²n². Show that cos(27nx) and sin(27nx) are both eigenfunctions belonging to eigenvalue A = 47²n² for each n = 1,2,... (in linear algebra language, this means that the eigenspaces are 2-dimensional!). Consider the eigenvalue problem y"(r) + Ay(x)= 0, y(0) = y(1), y'(0) = y(1). (a) (3 points) Show that there are no negative eigenvalues > < 0. (b) (3 points) Show that A = 0 is an eigenvalue with eigenfunction y = 1 (or, more generally, y = const # 0). (c) (4 points) Show that the positive eigenvalues are λ = 47²n². Show that cos(27nx) and sin(27nx) are both eigenfunctions belonging to eigenvalue A = 47²n² for each n = 1,2,... (in linear algebra language, this means that the eigenspaces are 2-dimensional!).