Modelling by differential equations

This paper aims to show the close relation between physics and mathematics taking into account especially the theory of differential equations. By analyzing the problems posed by the scientists in the 17 century, we note that physics is very important for the emergence of this theory. Taking into account this analysis, we show the characteristics of this relation both in current teaching and in the practices of first-year university students. We present the relation between the two disciplines in section 1. Section 2 deals with models and modelling. Having studied the emergence of differential equations in section 3, we focus on the status of modelling in current education in section 4. Lastly, we investigate the place of modelling in students’ practice and we finish by giving our conclusions in section 5. 1. Characteristics of the relations between mathematics and physics The relation between these two subjects comes within a philosophical and epistemological problem that we do not develop in this article. Nevertheless, this problem throws light on the possible relation between mathematics and physics as subjects in education. For a long time mathematicians and physicists have had various positions on this question. The statement that mathematics constitutes the language of physics is one of the explanations for the problem of the relations between these two disciplines. Poincare [1910] declares about this connection that all laws are drawn from experience, but in order to state them, a special language is needed; ordinary language is too poor and too vague to express relations so delicate, so meaningful. Thus the first reason why the physicist cannot do without mathematics is that it provides him with the only language that he can use. It is however essential to notice that this language carries out a reduction of the real system. Hence it is important to find a point of balance with the passage from system to the formula resting on the couple: experiment-theory, and phenomenon-laws. We will start by identifying the reciprocal relation, which allows mathematics to describe the physical phenomena and which allows physics to constitute an application field for mathematics. Of course, a physical phenomenon is not reduced to the mathematical concepts, but it is translated with the latter. So this point of view is in agreement with those who consider mathematics as a model of real phenomena, obtained by the process of modelling. 2. Modelling and models In this study, the term "modelling" indicates the translation of a real phenomenon to a model, and, especially, to a mathematical model. With respect to several didactic investigations Chevallard (1989), Dantal (2001) and Henry (2001) related to modelling, three main steps can be agreed: • Step 1: translation from reality to the model • Step 2: analysis of the model • Step 3: translation back to reality However, the above-mentioned scientists do not agree in the description of these steps. For example, Chevallard [1989, p. 53] described the task of modelling as follows: