Galerkin conditions

Define two sets of bases for $\psi(x) = \sum_i z_i \phi_i(x)$ and $u(x) = \sum_j w_j \phi_j(x)$. Then, we have $i = 1,...,m$ constraints:

$$\sum_{j=1}^m \left[\int_a^b c(x)\phi_i'(x)\phi_j'(x) dx + \int_a^b s(x)\phi_i(x)\phi_j(x)dx \right] = \int_a^b f(x)\phi_i(x)dx$$

Define matrices to represent the system of equations:

$$(\mathbf{K} + \mathbf{M})\mathbf{w} = \mathbf{f},$$ where $\mathbf{K}$ is the stiffness and $\mathbf{M}$ is the mass.