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Humans are curious, especially concerning the physical universe. Science began when (some) humans understood that ignorance exists, but they could do something about it within the context of a process that we now call the scientific methodology. Jointly with this methodology there arose an associated language, which we now call ‘mathematics’. One consequence of the construction of mathematical tools was their separate evolution into an intellectual field for the study of its own features, separate from its origins in science. Taking mathematics as the language of science, the precision study of the physical universe began with the construction of mathematical models for the analysis of particular physical phenomena. However, an interesting thing happened; essentially none of the associated mathematical equations could be solved exactly for their solutions. Consequently, a broad range of approximation techniques were created including computational methods for determining numerical solutions. Another, less used set of techniques, is based on acquiring only qualitative information on the desired solutions. While many of these procedures have been available since the early nineteenth century, many working scientists and researchers are not knowledgeable about their existence and/or how to apply them to the equations they hope to analyze, understand and solve. The main purpose of this small volume is to introduce some of these qualitative methods and show how they may be applied to differential equations. The reason for the focus on differential equations follows from the fact that many, if not most, of the physical systems have some form of mathematical model for which the mathematical structure is represented by differential equations.