In a recent course on Fourier Analysis and Approximation, the lecturer introduced the topic by attempting to solve a boundary value problem (BVP) $y''+y=\sin^3 x$, where $y$ is a function with a single argument.
The lecturer first solved the equation $y''+y=\lambda y$, which eventually leads to the line $\sum_k c_ky_k''+\sum_kc_ky_k=\sum_kc_k\lambda_ky_k$, where $k$ is the eigenvalue(s).
To me, the question was originally a boundary value problem, and the equation lecturer considered is an eigenvalue problem. While I understand that eigenvalue problems are important in BVP, I was a bit confused why we need to consider $y''+y=\lambda y$ in the first place (or why the question can be transformed into the eigenvalue problem). The lecturer briefly mentioned a name "Fourier method", but I failed to catch the details.
At a very basic level you need to look at the $\lambda=0$ eigenvalue problem just to know whether solutions to the BVP, if they exist, are unique. This part is necessary, it is not just a method that you can swap out for another method.
You can further use eigenfunctions to solve the whole problem if the forcing can be written as a superposition of eigenfunctions with nonzero eigenvalue. Suppose you can; that is, suppose the problem is $Ly=f$, $f=\sum a_k e_k$, $Le_k=\lambda_k e_k$ with nonzero $\lambda_k$. Then with $y=\sum b_k e_k$ the equation reads $Ly=\sum \lambda_k b_k e_k=\sum a_k e_k$, so you get a solution with $b_k=\frac{a_k}{\lambda_k}$. Then you have to add the general solution to the homogeneous BVP (which might or might not be just the zero function) to that to get the general solution.