This is the third of a sequence of articles devoted to the presentation of a direct method of analysis recently developed by one of the authors. The approach is quite general, since it is applicable to any linear operator, symmetric or nonsymmetric, regardless of its type. In particular, the theory includes steady-state and time-dependent problems. In this article the method is applied to ordinary differential equations, which constitute a very convenient illustration of its application because the analysis can be carried out in an exhaustive manner. General algorithms which yield the exact values of the solution or its derivatives are obtained in this manner.
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