Ordinary differential equations are the mathematical formulation for a great variety of problems in science and engineering, and frequently, two different problems are equivalent from a mathematical point of view when they are formulated by the same equations. Students acquire the knowledge of how to solve these equations (at least some types of them) using protocols and strict algorithms of mathematical calculation without thinking about the meaning of the equation. The aim of this work is that students learn to design network models or circuits in this way; with simple knowledge of them, students can establish the association of electric circuits and differential equations and their equivalences, from a formal point of view, that allows them to associate knowledge of two disciplines and promote the use of this interdisciplinary approach to address complex problems. Therefore, they learn to use a multidisciplinary tool that allows them to solve these kinds of equations, even students of first course of engineering, whatever the order, grade or type of non-linearity. This methodology has been implemented in numerous final degree projects in engineering and science, e.g., chemical engineering, building engineering, industrial engineering, mechanical engineering, architecture, etc. Applications are presented to illustrate the subject of this manuscript.

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The normalization of dimensionless groups that rule the system of nonlinear coupled ordinary differential equations defined by the Lotka–Volterra biological or chemical oscillator has been derived in this work by applying a normalized nondimensionalization protocol. The normalization procedure, which is quite accurate, does not require complex mathematical steps; however, a deep physical understanding of the problem is required to choose the appropriate references to define the dimensionless variables. From the dimensionless groups derived, the functional dependences of some unknowns of interest are established. Due to the coupled nature of the problem that induces temporal concentration rates of each species that are quite different at each point of the phase diagram, this diagram has been divided into four stretches corresponding to the four quadrants. For each stretch, the limit values (maximum or minimum) of the variables, as well as their duration, are expressed in terms of the dimensionless groups derived before. Finally, to check all the mentioned dependences, a numerical simulation has been carried out.

Classical dimensional analysis and nondimensionalization are assumed to be two similar approaches in the search for dimensionless groups. Both techniques, simplify the study of many problems. The first approach does not need to know the mathematical model, being sufficient a deep understanding of the physical phenomenon involved, while the second one begins with the governing equations and reduces them to their dimensionless form by simple mathematical manipulations. In this work, a formal protocol is proposed for applying the nondimensionalization process to ordinary differential equations, linear or not, leading to dimensionless normalized equations from which the resulting dimensionless groups have two inherent properties: In one hand, they are physically interpreted as balances between counteracting quantities in the problem, and on the other hand, they are of the order of magnitude unity. The solutions provided by nondimensionalization are more precise in every case than those from dimensional analysis, as it is illustrated by the applications studied in this work.

This paper presents a concise and orderly methodology to obtain universal solutions to different problems in science and engineering using the nondimensionalization of the governing equations of the physical–chemical problem posed. For its application, a deep knowledge of the problem is necessary since it will facilitate the adequate choice of the references necessary for its resolution. In addition, the application of the methodology to examples of coupled ordinary differential equations is shown, resulting in an interesting tool to teach postgraduate students in the branches of physics, mathematics, and engineering. The first example used for a system of coupled ordinary differential equations is a model of a continuous flow chemical reactor, where it is worth noting; on the one hand, the methodology used to choose the reference (characteristic) time and, on the other, the equivalence between the characteristic times obtained for each one of the species. The following universal curves are obtained, which are validated by comparing them with the results obtained by numerical simulation, where it stands out that the universal solution includes an unknown that must be previously obtained. The resolution of this unknown implies having a deep knowledge of the problem, a common characteristic when using the methodology proposed in this work for different engineering or physicochemical problems. Finally, the second example is a coupled oscillator, where it is worth noting that the appearance of characteristic periods that implicitly or explicitly affect the particles’ movement is striking.