Consider the eigenvalue problem

Question

Question:

Eigenvalue proofs

Consider the eigenvalue problem

d²y/dx² + λy(x) = 0      0

with boundary conditions y(0)=0,       y(1)+y'(1)=0

- Determine the general solution

- Apply the boundary conditions to solve for the eigenvalues

- Show that

a. The eigenvalues are all positive,

b. There are a countably infinite number of eigenvalues,

c. All the eigenvalues are simple roots.

d. The eigenvalue λ_n corresponding to the nth eigenfunction tends to ∞  n → ∞

4. Find an approximate expression for λ_n as n → ∞