Consider the boundary problem $$u''+k^2u=0 \qquad (1)\\ u(0)=u_l, u(1)=u_r \qquad\\ \text{domain } \Omega = (0,1)\qquad$$
When I discretize this problem using the Finite Element Method I get the following linear system:
$$(k^2M-K)u=b \qquad(2)$$
where the matrix $k^2M-K$ can be positive or negative definite, positive or negative semidefinite or indefinite. This means that $(2)$, depending on $k$, may have a unique solution, no solutions or infinitely many solutions.
However, the analytic solution of $(1)$ has the unique solution $$u(x) = \sin(k x) [u_r \csc(k) - u_l \cot(k)] + u_l \cos(k x)\qquad (3)$$ which is a unique solution. Of course $k$ may vary, but $k$ is assumed to be a constant, so when we solve $(1)$ we fix our $k$.
Yet I was told that $(1)$ is supposed to have infinitely many solutions as well. Where am I being wrong?