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## Homework Statement

Given

X''(x) + lambda*X(x) = 0

X(0) = X'(0), X(pi) = X'(pi)

Find all eigenvalues and eigenfunctions.

## Homework Equations

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)?