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A unified approach, named discretized boundary equation (DBE) method, is introduced for two-dimensional (2-D) scattering problems. It is based on the discretization of field expressions for one or two components of the scattered field. The DBEs can be used either on the object surface to obtain the solution directly or on the truncation boundary of a finite difference (FD) or finite element (FE) mesh as termination conditions. This paper describes the general theory of the DBE method and key points or limitations for its implementation. A new on-surface formulation for the solution of scattering by perfectly conducting cylinders is presented as an application of the two-component version of the DBE method and validated through numerical examples. Mesh termination conditions for the FD or FE method are derived based on the one-component formulation of the DBE method and their equivalence and difference to the measured equation of invariance are discussed. In particular, the DBE obtained with the minimum norm least squares solution is investigated thoroughly and its validity and features are demonstrated through numerical results, generated together with the FD method, for scattering by cylinders with various material properties.