A method of determining coefficients α_k in a power series solution y(x) = y_0(x) + sum_(k = 1)^n α_k y_k(x) of the ordinary differential equation L^~[y(x)] = 0 so that L^~[y(x)], the result of applying the ordinary differential operator to y(x), is orthogonal to every y_k(x) for k = 1, ..., n (Itô 1980). Galerkin methods are equally ubiquitous in the solution of partial differential equations, and in fact form the basis for the finite element method.

finite element method