X''(x) + lambda*X(x) = 0
X(0) = X'(0), X(pi) = X'(pi)
Find all eigenvalues and eigenfunctions.
Case lambda = 0
Case lambda > 0
Case lambda < 0
The Attempt at a Solution
First case, X(x) = Ax + B but the function doesn't satisfy the boundary condition at pi.
Second case, lambda = k^2
X(x) = Acos(kx) + Bsin(kx)
X'(x) = Bkcos(kx) - Aksin(kx)
X(pi) - X'(pi) = Bsin(kpi) + Bk^2 sin(kpi)
=> 1+ k^2 = 0 so lambda = -1
a) I was told the eigenfunction for that is X(x) = Ce^x. How?
b) Also, is that just the lambda_0 or does it satisfy all possibly eigenfunctions (i.e. eigenvalue lambda_n)?