I want to solve the following system of equations. It can be seen as a double Fredholm integral equation,
$\begin{multline*} \displaystyle\int f(x)(\cos a_nx - \cos b_nx)dx = C_n-D_n\\ + \displaystyle\sum_{i=0}^N \left( C_i \displaystyle\int \cos a_nx \cos a_ixdx - D_i \displaystyle\int \cos a_nx \cos b_ixdx \right) \\ + \displaystyle\sum_{j=0}^N \left(C_j \displaystyle\int \cos b_nx \cos a_jxdx - D_j \displaystyle\int \cos b_nx \cos b_jxdx\right) \end{multline*} $
so the system of equations should be $(2N+2)^2$. but in reality I can't see how the system looks explicitly. In addition, it seems natural that many integrals are going to be cancelled, but since I don't have more information about $a_n$ and $b_n$, I can't be so bold as to delete them.
Thank you very much if you can give me at least some more general ideas of how this system should behave.