A simple question, hopefully - consider, for example, the standard BVP defined as follows
$$y^{\prime\prime} + \lambda y = 0\qquad y\left(0\right) = 0 \qquad y\left(2\pi\right) = 0$$
The traditional approach here is to find the eigenvalues and associated eigenfunctions, which would be something like
$$\lambda_n = \dfrac{n^2}{4} \qquad y_n\left(x\right) = \sin\left(\dfrac{n x}{2}\right)$$
Now, this might not be the best question to ask but following this procedure, what exactly will be the final "solution" to the original problem? Will it be a non-trivial linear combination of these eigenfunctions (how do we account for the different $\lambda_n$ in this case) or just the individual eigenfunctions themselves?