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

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