The Galerkin method

Collocation = differentiation, Galerkin = integration

Weak Solution: if $u(x)$ is a function that satisfies the weak form for all valid $\psi(x)$, then $u$ is a weak solution

$$\int_a^b \left[c(x)u'(x)\psi'(x) + s(x)u(x)\psi(x)\right]dx = \int_a^b f(x)\psi(x)dx.$$