For the $-\Delta$ operator is positive definite if the Dirichlet boundary is non-empty, here your problem is actually to find $\lambda >0$ such that:
$$
-X'' = \lambda X.\tag{1}
$$
General solution to (1) is
$$
X = A\cos(\sqrt{\lambda}x)+ B\sin(\sqrt{\lambda}x),
$$
and
$$
X' = -A\sqrt{\lambda}\sin(\sqrt{\lambda}x)+ B\sqrt{\lambda}\cos(\sqrt{\lambda}x).
$$
The boundary condition
$$
X'(0)=0\implies B = 0,
$$
and
$$
X(L) = a\implies A\cos(\sqrt{\lambda}L) = a\implies A = \frac{a}{\cos(\sqrt{\lambda}L)}.\tag{2}
$$
Therefore the solution is:
$$
X_{\lambda}(x) = \frac{a\cos(\sqrt{\lambda}x)}{\cos(\sqrt{\lambda}L)}.
$$
Given the denominator is non-zero.
Thanks to the remark of Michael Seifert. This problem with a non-homogeneous boundary condition is like a Helmholtz equation, in order it has a unique solution $\lambda$ CANNOT be an eigenvalue of the eigenvalue problem below: i.e., for $j=1,2,3,\dots$
$$
\lambda \neq \frac{(2j - 1)^2 \pi^2}{4 L^2}
$$
in order that (1) has a unique solution with boundary value
$$
X'(0) = 0, \quad X(L)=a.
$$
Notice that for Dirichlet eigenvalue problem on an interval, you could only have complete squares $k^2$ (un-normalized) as $\lambda$. The wikipedia pages mellow posted has normalized eigenfunctions for mixed Neumann-Dirichlet boundary value problems as:
$$
X_j(x) = \sqrt{\frac{2}{L}} \cos\left(\frac{(2j - 1) \pi x}{2 L}\right),
$$
with eigenvalue
$$
\lambda_j = \frac{(2j - 1)^2 \pi^2}{4 L^2}.
$$
This is because the wikipedia page uses homogeneous boundary condtions on both Neumann and Dirichlet boundaries:
$$
X'(0) = 0\quad \text{and}\quad X(L) = 0.
$$
Notice the second Dirichlet boundary condition will change (2) to
$$
A\cos(\sqrt{\lambda}L) = 0\implies \sqrt{\lambda}L = \frac{(2j-1)\pi}{2}.
$$
That's why wikipedia's page has that solution.
